Ralf Hiptmair and Ronald H.W. Hoppe 
Mixed Finite Element Discretization of Continuity Equations arising inSemiconductor Device Simulation 
Proceedings of a Conference held at the Mathematisches Forschungsinstitut
Oberwolfach, July 511, 1992, R. Bank, R. Bulirsch, H. Gajewski, and K.
Merten, eds., 1993, pp. 197217
Also:Technical Report TUMM9302, Mathematisches Institut, Technische Universität München 
Keywords :convectiondiffusion problem, flux oriented schemes, hybridization, Lagrangian multipliers,mixed finite elements, RaviartThomas elements 
Abstract: In the wake of decoupling and linearization semiconductor device simulation based on van Roosbroecks's equations requires the solution of convectiondiffusion equations. It is well known that due to the occurrence of local regions of strong convection standard discretizations do not behave properly. As an alternative among others, mixed methods have been suggested having their roots in the dual variational formulation of the convectiondiffusion problem.Their efficient implementation has to make use of Lagrangian multipliers. In a novel approach we already introduce the multiplier prior to discretizing, through a process called hybridization. In the sequel we use the resulting variational problem to develop a new discretization scheme. Next, we outline how to implement a standard mixed scheme and investigate some of its aspects. Finally, the behaviour of the mixed method is illustrated by a series of numerical experiments. 
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Ralf Hiptmair 
Nonconforming Mixed Discretization of Second Order Elliptic Problems 
Technical Report TUMM9404, Math. Inst. TU München, 1994 
Keywords : mixed finite elements, nonconforming finite elements, patch test, RaviartThomas spaces 
Abstract: In the realm of finite element discretizations of elliptic problems a basic distinction can be made between conforming and nonconforming methods. The latter rely on approximations not contained in the space the exact solution lives in. Since additional difficulties are thus introduced nonconforming approaches usually are a last resort: if satisfactory conforming spaces prove elusive they are given a try. For primal variational problems a mature theory of nonconforming methods has already been developed. This paper seeks to extend these ideas to the mixed formulation of second order boundary value problems. A ``patchtest'' is designed as a powerful tool for probing the viability of nonconforming approximations. In particular the criteria are utterly problemindependent. Consequently they can dispense with any regularity requirements. 
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Ralf Hiptmair and Ingrid Schmitt 
Compatibility Conditions for Semiconductor Equations 
Technical Report TUMM9409, Math. Inst. TU München, 1994 
Keywords : semiconductor equations, nonlinear elliptic systems, ScharfetterGummel scheme, BaligaPatankar scheme, divergence free upwinding 
Abstract: The basic driftdiffusion model for semiconductors gives rise to a system of three coupled partial differential equations: one potential eqation and two continuity equations. G\ärtner was the first to point out that, using different discretization schemes for both types of equations, the coupling might not be represented correctly. This paper further elaborates his ideas: in an abstract setting we explain how a dicretization scheme should take into account the interaction of the equations. To that end we formulate two {\em compatibility conditions}. Then we develop a technique for quickly assessing the viability of a particular scheme. It relies on purely local examinations and a few heuristic rules. Next we investigate several existing schemes to what extent they meet the compatibility conditions. We show that only the classical ScharfetterGummel scheme passes the tests. All other methods under examination are rejected. 
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Ralf Hiptmair, Thomas Schiekofer and Barbara Wohlmuth 
Multilevel Preconditioned Augmented Lagrangian Techniques for 2nd Order Mixed Problems 
Computing, 57 (1996), pp. 2548 Also: Report No. 328, Math.Nat. Fakultät, Univ. Augsburg, 1995 
Keywords : Augmented Lagrangian algorithm, saddle point problems, mixed finite elements, multilevel preconditioners 
Abstract: We are concerned with the efficient solution of saddle point problems arising from the mixed discretization of 2nd order elliptic problems in two dimensions. We consider the mixed discretization of the boundary value problem by means of lowest order RaviartThomas elements. This leads to a saddle point problem, which can be tackled by Uzawalike iterative solvers. We suggest a prior modification of the saddle point problem according to the augmented Lagrangian approach in order to make it more amenable to the iterative procedure. In order to boost the speed of iterative methods, we additionally employ a multilevel preconditioner first presented by Vassilevski and Wang in. It is based on a special splitting of the space of vector valued fluxes, which exploits the close relationship between piecewise linear continuous finite element functions and divergence free fluxes. We prove that this splitting gives rise to an optimal preconditioner: it achieves condition numbers bounded independently on the depth of refinement. The proof is set in the framework of Schwarz methods. It relies on established results about standard multilevel methods as well as a strengthened CauchySchwarz inequality for lowest order RaviartThomas spaces. 
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Ralf Hiptmair, Ronald H.W. Hoppe and Barbara Wohlmuth 
Coupling problems in microelectronic device simulation 
Proc. of the 11th GAMM Sem. Kiel, January 1995, Notes on Numerical Fluid Mechanics Vol. 51, p. 8695, (Eds.: W. Hackbusch and G. Wittum), Vieweg, Braunschweig 
Keywords :semiconductor simulation, mixed finite elements 
Abstract: The costeffective design of electronic microstructures requires an advanced modeling and coupled simulation of various physical effects. The classical isothermal approach leads to the basic driftdiffusion model for semiconductor device simulation. In the stationary case, it represents a coupled nonlinear system consisting of a Poisson equation for the electricpotential and two continuity equations for the electron and hole flow. We discuss various discretization schemes with special emphasis on mixed finite element methods and we further address efficient numerical solution techniques including adaptive multilevel methods. Finally, to allow for ambient conditions such as external magnetic fields we consider consistent extensions of the classical model and discuss perspectives for their numerical treatment. 
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Ralf Hiptmair 
Concepts for an Object Oriented Finite Element Code 
Report No. 335, Math.Nat. Fakultät, Uni Augsburg, 1995 
Keywords : Object oriented programming, adaptive finite element, multilevel schemes 
Abstract: Adaptive multilevel algorithms along with finite element
discretizations are the most po werful tools for the fast and accurate solution of
elliptic boundary value problems over irregular do mains. They involve the delicately
tuned interaction of a posteriori error estimation, local mesh refinement, multilevel
computations and iterative solution. This introduces unprecedented complexity into mathematical software, in terms of both sheer size of the code and sophisticated dynamic data structures. Advanced techniques of software engineering have to be used in order to develop reliable codes quickly. In particular, we relied on the object oriented paradigm of software design. It requires us to split the task into logical units and to fix several levels of abstraction within those units. This leads to a set of socalled classes, arranged in a tree, which jointly manage data and supply functions. Crudely speaking, two types of classes could be identified: Local classes and global classes. The former are concerned with components of the mesh and the latter deal with the entire mesh. Thus highly versatile building blocks for adaptive multilevel finite element schemes have been created. They relieve the user of caring about the internal management of the grid and numerical data and offer well defined interfaces. They can be quickly assembled into a code for a vast range of boundary value problems, including systems of equations and saddle point problems. 
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Ralf Hiptmair and Ronald H.W. Hoppe 
Multilevel computation of magnetic fields 
Report No. 342, Math.Nat. Fakultät, Uni Augsburg, 1995 
Keywords : Maxwell's equations, Finite elements, Multilevel techniques, Nedelec's spaces, Multilevel preconditioning 
Abstract: We are concerned with the efficient computation of the magnetic field induced by a stationary strictly conservative current. According to Maxwell's equations the magnetic field has vanishing divergence and its $\curl$ equals the current. The current is assumed to be discretized by means of lowest order RaviartThomas elements. The magnetic field is sought in N'ed'elec's H(curl)conforming finite element spaces of order 1. We suggest a multilevel approach for the solution of this problem. It involves two steps: First a discrete stream function of the current is determined. This is a direct procedure based on a multilevel splitting of the current. In a second step a curlfree correction is computed by means of a multilevel preconditioned CG method to ensure the divergence free condition. The overall amount of work is proportional to the number of unknown values of the discrete magnetic field. 
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Ralf Hiptmair 
Multilevel Preconditioning for Mixed Problems in Three Dimensions 
PhD Thesis, Vol. 8 of Augsburger mathematischnaturwissenschaftliche Schriften, Verlag Wißner, Augsburg, 1996 
Keywords : 
Abstract: This work aims to provide a rigorous theoretical examination of multilevel preconditioning schemes for some discrete variational problems in three dimensions, which involve the differential operators div and curl. Such kinds of problems occur, for instance, in the dual formulation of elliptic boundary value problems, in which case they are posed over subspaces of H(div) and h(curl). The investigations are set in a finite element framework relying on simplicial meshes and the finite element schemes introduced by Raviart and Thomas. We take a fresh look at the construction of these spaces, viewing them from the angle of differential forms. Thus, we arrive at a fairly canonical procedure to obtain these particular finite elements and forge a unified analysis of their properties. In addition, we managed to establish approximation estimates in fractional Sobolev spaces and new discrete extension theorems. Our main focus is on the augmented Lagrangian technique , applied to the saddle point system arising from the mixed finite element discretization of an ordinary scalar second order elliptic problem. Uzawa algorithms, the minimal residual method and the method of Bank, Welfert and Yserentant provide algorithms for its iterative solution. We show that it takes only an efficient preconditioner for discrete operators related to the bilinear form (u,v)+r(divu,divv), to achieve methods of optimal computational complexity. Of course, the preconditioner must not be adversely affected by large values of the augmented Lagrangian parameter r. We extend the approach of Vassilevski and Wang in two dimensions to obtain a multilevel splitting of RaviartThomas spaces in 3D, built upon a sequence of nested triangulations. For the treatment of the crucial divergence free vector fields we resort to a nodal BPXtype decomposition of N'ed'elec spaces. We discover that with slight modifications the hierarchical bases scheme of Cai, Goldstein and Pasciak is covered, as well. Based on algebraic multilevel theory , we investigate the stability of the decompositions of lowest order N'ed'elec spaces with respect to the bilinear form (curl u,curl v). To cope with its nontrivial kernel we switch to the related quotient space. Under certain assumptions on the regularity of a curl curlboundary value problem, duality arguments according to Zhang yield the stability of the nodal splitting, independent on the number of refinement levels. By the extension theorem this result carries over to general domain if no boundary values are imposed. The findings also show that the direct elimination of the nonsolenoidal part of the flux suggested by Ewing and Wang for 2D applications is just as efficient for mixed problems in 3D. Also they enable us to construct fast solvers for first order system least squares discretizations of second order elliptic problems. Moreover, the results prove useful for designing multilevel schemes for the computation of vector potentials in magnetostatics. 
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Ralf Hiptmair and Ronald H.W. Hoppe 
Multilevel Methods for Mixed Finite Elements in Three Dimensions 
Numerische Mathematik 82 (1999) 2, 253279 Also: Report No. 359, Math.Nat. Fakultät, Univ. Augsburg, October 1996 
Keywords : Mixed finite elements, multilevel methods, RaviartThomas finite element, Nedelec finite elements, H(curl), H(div) 
Abstract: In this paper we consider second order scalar elliptic boundary value problems posed over threedimensional domains and their discretization by means of mixed RaviartThomas finite elements. This leads to saddlepoint problems featuring a discrete flux vector field as additional unknown. Following Ewing and Wang, the proposed solution procedure is based splitting the flux into divergence free components and a remainder. It leads to variational problem involving solenoidal RaviartThomas vector fields. A fast iterative solution method for this problem is presented. It exploits the representation of divergence free vector fields as curls of the H(curl)conforming finite element functions introduced by N'ed'elec. We show that a nodal multilevel splitting of these finite element spaces gives rise to an optimal preconditioner for the solenoidal variational problem: Duality techniques in quotient spaces and modern algebraic multigrid theory are the main tools for the proof. 
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Ralf Hiptmair 
Canonical Construction of Finite Elements 
Mathematics of Computation, 68 (1999), pp. 13251346 Also: Report No. 360, Math.Nat. Fakultät, Univ. Augsburg, November 1996 
Keywords : Finite elements, differential forms, RaviartThomas elements, Nedelec elements, Whitney forms, discrete potentials 
Abstract: The mixed variational formulation of many elliptic boundary value problems involves vector valued function spaces, like, in three dimensions, H(curl) and H(div). Thus finite element subspaces of these function spaces are indispensable for effective finite element discretization schemes. Given a simplicial triangulation of the computational domain, among others, Raviart, Thomas and N'ed'elec have found suitable conforming finite elements for H(div) and H(curl), respectively. At first glance, it is hard to detect a common guiding principle behind these approaches. We take a fresh look at the construction of the finite spaces viewing them from the angle of differential forms. This is motivated by the wellknown relationships between differential forms and differential operators: both div, (curl and grad can be regarded as special incarnations of the exterior derivative of a differential form. Moreover, in the realm of differential forms most concepts are basically dimensionindependent. Thus, we arrive at a fairly canonical procedure to construct conforming finite element subspaces of function spaces related to differential forms. In any dimension we can give a simple characterization of the local polynomial spaces and degrees of freedom underlying the definition of the finite element spaces. With unprecedented ease we can recover the familiar H(div) and H(curl)conforming finite elements and establish the unisolvence of degrees of freedom. In addition, the use of differential forms makes it possible to establish crucial algebraic properties of the canonical interpolation operators and representation theorems in a single sweep for all kinds of spaces. 
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Rudolf Beck and Ralf Hiptmair 
Multilevel Solution of the TimeHarmonic Maxwell Equations Based on Edge Elements 
International Journal of Numerical Methods in Engineering, 45 (1999), pp. 901920 Also: Report SC9651, ZIBBerlin, 1996 
Keywords : maxwell's equations, multigrid, waveguide computation,edge elements, a posteriori error estimation 
Abstract: A widely used approach for the computation of timeharmonic electromagnetic fields is based on the wellknown doublecurl equation for either E or H. An appealing choice for finite element discretization are edge elements, the lowest order variant of a H(curl)conforming family of finite elements. We end up with a large sparse linear system of equations, which can only be solved iteratively. In this paper we focus on fast multilevel preconditioned iterative schemes. Yet, the nullspace of the curloperator comprises a considerable part of all spectral modes on the finite element grid. Thus standard multilevel solvers are rendered inefficient, as they essentially hinge on smoothing procedures like GaussSeidel relaxation, which cannot provide a satisfactory error reduction for modes with very small or even negative eigenvalues. A remedy is offered by an extended multilevel algorithm that relies on corrections in the space of discrete scalar potentials. After every standard Vcycle with respect to the canonical basis of edge elements, error components in the nullspace are removed by an additional projection step. Furthermore, a simple criterion for the coarsest mesh is derived to guarantee both stability and efficiency of the iterative multilevel solver. For the whole scheme we observe convergence rates independent of the refinement level of the mesh. The sequence of nested meshes required for multilevel techniques is constructed by adaptive refinement. For controlling adaptive mesh refinement we have devised an aposteriori error indicator based on stress recovery. } 
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Ralf Hiptmair 
Multigrid Method for H(div) in Three Dimensions 
Electronic Transactions on Numerical Analysis, 6 (1997), pp. 777 Also: Report No. 368, Math.Nat. Fakultät, Univ. Augsburg, May 1997 
Keywords : Multigrid, RaviartThomas finite elements, N\'ed\'elec's finite elements, multilevel, mixed finite elements 
Abstract: We are concerned with the design and analysis of a multigrid algorithm for H(div)elliptic linear variational problems. The discretization is based on H(div)conforming RaviartThomas elements. A thorough examination of the relevant bilinear form reveals that a separate treatment of vector fields in the kernel of the divergence operator and its complement is paramount. We exploit the representation of discrete solenoidal vector fields as $\curl$s of finite element functions in socalled N'ed'elec spaces. It turns out that a combined nodal multilevel decomposition of both the RaviartThomas and N'ed'elec finite element spaces provides the foundation for a viable multigrid method. Its GaußSeidel smoother involves an extra stage where solenoidal error components are tackled. By means of elaborate duality techniques we can show the asymptotic optimality in the case of uniform refinement. Numerical experiments confirm that the typical multigrid efficiency is actually achieved for model problems. 
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Ralf Hiptmair 
Multigrid Method for Maxwell's Equations 
SIAM Journal of Numerical Analysis, Vol. 36, No. 1, 1999, pp. 204225 Also: Report No. 374, Math.Nat. Fakultät Univ. Augsburg, August 1997 
Keywords : Maxwell's equations, N\'ed\'elec's finite elements, edge elements, multilevel methods, multigrid 
Abstract: In this paper we are concerned with the efficient solution of discrete variational problems related to the bilinear form (curl.,curl.)+(.,.) defined on H(curl). This is a core task in the timedomain simulation of electromagnetic fields, if implicit timestepping is employed. We rely on N'ed'elec's H(curl)conforming finite elements (edge elements) to discretize the problem. We construct a multigrid method for the fast iterative solution of the resulting linear system of equations. Since proper ellipticity of the bilinear form is confined to the complement of the kernel of the curloperator, Helmholtzdecompositions are the key to the design of the algorithm: Kern(curl) and its complement require a separate treatment. Both can be tackled by nodal multilevel decompositions where for the former the splitting is set in the space of discrete scalar potentials. Under certain assumptions on the computational domain and the material functions a rigorous proof of the asymptotic optimality of the multigrid method can be given; it shows that convergence does not deteriorate on very fine grids. The results of numerical exeriments confirm the practical efficiency of the method. 
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Ralf Hiptmair 
Nonconforming Vector Valued Finite Elements 
EastWest J.~Num.~Math., 5(3), pp. 163182, 1997 Also: Report No. 375, Math.Nat. Fakultät, Univ. Augsburg, August 1997 
Keywords : Nonconforming finite elements, RaviartThomas finite elements, N\'ed\'elec's finite elements, mixed finite elements, generalized patch test, differential forms 
Abstract: In the realm of finite element discretizations of elliptic problems a basic distinction can be made between conforming and nonconforming methods. The latter rely on approximations not contained in the spaces in which the continuous variational problem is posed. This paper investigates nonconforming finite element approximations of the spaces H(div) and H(curl) of vector valued functions. First, we extend the "generalized patch test" which has been developed to provide necessary and sufficient conditions for the viability of nonconforming schemes for standard Sobolev spaces. Then we use the calculus of differential forms to derive coupling conditions sufficient for success in the patch test. Based on this result we design convergent nonconforming methods for a number of variational problems involving the abovementioned function spaces. Finally, we point out that a naive approach leads to an inconsistent scheme. 
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Ralf Hiptmair and Andrea Toselli 
Overlapping Schwarz methods for vector valued elliptic problems in three dimensions 
in "Parallel solution of PDEs", IMA Volume in Mathematics and
its Applications, Springer, BerlinHeidelbergNew York, 1998 Also: Report TR1997746, Courant Institute, New York, 1997 
Keywords : Schwarz methods, domain decomposition, multilevel methods, multigrid, RaviartThomas finite elements, N\'ed\'elec's finite elements 
Abstract: This paper is intended as a survey of current results on algorithmic and theoretical aspects of overlapping Schwarz methods for discrete H(curl and H(div)elliptic problems set in suitable finite element spaces. The emphasis is on a unified framework for the motivation and theoretical study of the various approaches developed in recent years. Generalized Helmholtz decompositions  orthogonal decompositions into the null space of the relevant differential operator and its complement  are crucial in our considerations. It turns out that the decompositions the Schwarz methods are based upon have to be designed separately for both components. In the case of the null space, the construction has to rely on liftings into spaces of discrete potentials. Taking the cue from wellknown Schwarz schemes for second order elliptic problems, we devise uniformly stable splittings of both parts of the Helmholtz decomposition. They immediately give rise to powerful preconditioners and iterative solvers. 
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Peter Deuflhard, Ralf Hiptmair, Ronald H.W. Hoppe and Barbara Wohlmuth 
Adaptive Multilevel Method for Edge Element Discretizations of Maxwell's Equations 
Surveys on Mathematics for Industry, 8 (1998), pp. 271312 Also: Technical Report, SC9766, ZIBBerlin, 1997 
Keywords : Maxwell's equations, edge elements, N\'ed\'elec's elements, multigrid methods, a posteriori error estimators, waveguide simulation 
Abstract: The focus of this paper is on the efficient solution of boundary value problems involving the double curloperator. Those arise in the computation of electromagnetic fields in various settings, for instance when solving the electric or magnetic wave equation with implicit timestepping, when tackling timeharmonic problems or in the context of eddycurrent computations. Their discretization is based on N'ed'elec's curlconforming edge elements on unstructured grids. In order to capture local effects and to guarantee a prescribed accuracy of the approximate solution adaptive refinement of the grid controlled by a posteriori error estimators is employed. The hierarchy of meshes created through adaptive refinement forms the foundation for the fast iterative solution of the resulting linear systems by a multigrid method. The guiding principle underlying the design of both the error estimators and the multigrid method is the separate treatment of the kernel of the curloperator and its orthogonal complement. Only on the latter we have proper ellipticity of the problem. Yet, exploiting the existence of computationally available discrete potentials for edge element spaces, we can switch to an elliptic problem in potential space to deal with nullspace of $\curl$. Thus both cases become amenable to strategies of error estimation and multigrid solution developed for second order elliptic problems. The efficacy of the approach is confirmed by numerical experiments which cover several model problems and an application to waveguide simulation. 
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Ralf Hiptmair, Ronald H.W. Hoppe and Barbara Wohlmuth 
Residual Based A Posteriori Error Estimators For Eddy Current Computation 
Report 112, SFB 382, Universität Tübingen, March 1999 
Keywords : Residual based a posteriori error estimation, Nedelec's edge elements, Helmholtz decomposition, eddy currents 
Abstract: We consider H(curl)elliptic problems that have been discretized by means of N'ed'elec's edge elements on tetrahedral meshes. Such problems occur in the numerical compuation of eddy currents. From the defect equation we derive localized expressions that can be used as a posteriori error estimators to control adaptive refinement. Under certain assumptions on material parameters and computational domains, we derive local lower bounds and a global upper bound for the total error measured in the energy norm. The fundamental tool in the numerical analysis is a Helmholtztype decomposition of the error into an irrotational part and a weakly solenoidal part. 
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Vasile Gradinaru and Ralf Hiptmair 
Whitney Elements on Pyramids 
ETNA, 8 (1999), pp. 154168 Also: Report 113, SFB 382, Universität Tübingen, March 1999 
Keywords : Whitney elements, edge elements, pyramidal element 
Abstract: Conforming finite elements in H(div) and H(curl), can be regarded as discrete differential forms (Whitneyforms). The construction of such forms is based on an interpolation idea, which boils down to a simple extension of the differential form to the interior of elements. This flexible approach can accommodate elements of more complicated shapes than merely tetrahedra and bricks. The pyramid serves as an example for the successful application of the construction: New Whitney forms are derived for it. 
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Bernd Engelmann, Ralf Hiptmair, Ronald H.W.Hoppe, and George Mazurkevitch 
Numerical Simulation of Electrorheological Fluids Based on an Extended Bingham Model 
Technical Rreport 99/2, SFB 342, TUMünchen, Universität Augsburg, January 1999 (to appear in "Computing and Visualization in Science") 
Keywords : Bingham fluid, augmented Lagrangians, finite elements, iterative solver 
Abstract: In the context of macroscopic simulation of electrorheological fluids, we present an extension of the classical Bingham model. It accommodates arrangements beyond pure shear flows and will make possible fully threedimensional simulations. For the numerical solution of the resulting nonsmooth minimization problem we propose the method of augmented Lagrangians which turns out to be an appropriate iterative solver for such problems. Finally, we present computational results illustrating the electrorheological effect for various shear rates and electric field strengths in the case of an electrorheological suspension rotating between two revolving cylinders. 
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R. Hiptmair 
Multilevel Gauging for Edge Elements 
Report No. 121, SFB 382, Universität Tübingen, May 1999. To appear in Computing 
Keywords : Computational electromagnetism, edge elements, hierarchical bases, multilevel decomposition, vector potentials, gauging. 
Abstract:
The vector potential of a solenoidal vector field, if it exists, is not unique in
general. Any procedure that aims to determine such a vector potential typically
involves a decision on how to fix it. This is referred to by the term gauging.
Gauging is an important issue in computational electromagnetism, whenever discrete
vector potentials have to be computed. In this paper a new gauging algorithm for
discrete vector potentials is introduced that relies on a hierarchical multilevel
decomposition. With minimum computational effort it yields vector potentials whose
L 
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S. Börm and R. Hiptmair 
Analysis of Tensor Product Multigrid 
Numerical Algorithms 26, 219234, 2001 Also: Report No. 123, SFB 382, Universität Tübingen, May 1999 
Keywords : Robust multigrid methods, anisotropic elliptic problems, semicoarsening. 
Abstract: We consider anisotropic second order elliptic boundary value problems in two dimensions, for which the anisotropy is exactly aligned with the coordinate axes. This includes cases where the operator features a singular perturbation in one coordinate direction, whereas its restriction to the other direction remains neatly elliptic. Most prominently, such a situation arises when polar coordinates are introduced. The common multigrid approach to such problems relies on line relaxation in the direction of the singular perturbation combined with semicoarsening in the other direction. Taking the idea from classical Fourier analysis of multigrid, we employ eigenspace techniques to separate the coordinate directions. Thus, convergence of the multigrid method can be examined by looking at onedimensional operators only. In a tensor product Galerkin setting, this makes it possible to confirm that the convergence rates of the multigrid Vcycle are bounded independently of the number of grid levels involved. In addition, the estimates reveal that convergence is also robust with respect to a singular perturbation in one coordinate direction. Finally, we supply numerical evidence that the algorithm performs satisfactorily in settings more general than those covered by the proof. 
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R. Hiptmair 
Discrete Hodge operators 
Numerische Mathematik ,  , 2001 also: Report No. 126, SFB 382, Universität Tübingen, October 1999 
Keywords : Discrete differential forms, discrete vector analysis, finite elements, finite volume methods, error estimates 
Abstract: Many linear boundary value problems arising in computational physics can be formulated in the calculus of differential forms. Discrete differential forms provide a natural and canonical approach to their discretization. However, much freedom remains concerning the choice of discrete Hodge operators, that is, discrete analogues of constitutive laws. A generic discrete Hodge operator is introduced and it turns out that most finite element and finite volume schemes emerge as its specializations. We reap the possibility of a unified convergence analysis in the framework of discrete exterior calculus. 
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R. Beck, R. Hiptmair, and B. Wohlmuth 
Hierarchical Error Estimator for Eddy Current Computation 
ENUMATH 99  Proceedings of the 3rd European Conference on Numerical Mathematics and Advanced Applications, Jyväskylä, Finland, July 2630, P. Neittaanmäki and T. Tiihonen, eds., World Scientific, Singapore (2000), pp. 110120 Also: Report No. 127, SFB 382, Universität Tübingen, October 1999 
Keywords : Edge elements, a posteriori error estimator, hierarchical error estimator 
Abstract: We consider the quasimagnetostatic eddy current problem discretized by means of lowest order curlconforming finite elements (edge elements) on tetrahedral meshes. Bounds for the discretization error in the finite element solution are desirable to control adaptive mesh refinement. We propose a local aposteriori error estimator based on higher order edge elements: The residual equation is approximately solved in the space of phierarchical surpluses. Provided that a saturation assumption holds, we show that the estimator is both reliable and efficient. 
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S. Börm and R. Hiptmair 
Multigrid Computation of Axisymmetric Electromagnetic Fields 
Adv. Comp. Math., 16(4), 2001, pp. 331356. also Report No. 138, SFB 382, Universität Tübingen, February 2000 
Keywords : Computational electromagnetism, multigrid, finite elements, edge elements, semicoarsening, cylindrical symmetry, degenerate problems 
Abstract: The focus of this paper is on boundary value problems for Maxwell's equations that feature cylindrical symmetry both of the domain and the data. Thus, by resorting to cylindrical coordinates, a reduction to two dimensions is possible. However, cylindrical coordinates introduce a potentially malicious singularity at the axis rendering the variational problems degenerate. Ultimately, this cripples the performance of a standard multigrid solver. Line relaxation in radial direction and semicoarsening can successfully reign in the degeneracy. In addition, the lack of strong ellipticity of the doublecurl operator entails using special hybrid smoothing procedures. All these techniques combined yield a fast multigrid solver. The theoretical investigation of the method relies on blending generalized Fourier techniques and modern variational multigrid theory. We first determine invariant subspaces of the multigrid iteration operator and analyze the smoothers therein. Under certain assumptions on the material parameters we manage to show uniform convergence of a symmetric Vcycle. 
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R. Hiptmair 
Symmetric Coupling for Eddy Current Problems 
Report No. 148, SFB 382, Universität Tübingen, March 2000. Submitted to SIAM J. Numer. Anal. 
Keywords : Eddy current problem, trace spaces, boundary integral operators, symmetric coupling, edge elements, boundary elements. 
Abstract: In this paper a novel symmetric FEMBEMcoupling for the Ebased eddy current model is derived in a rigorous fashion. To that end the properties of potentials and boundary integral operators arising from a StrattonChutype representation formula for the electric field in the nonconducting region are thoroughly analyzed in a Hilbertspace setting. It yields a variational problem with symmetric bilinear form that is coercive in the natural function spaces. Unknowns are the electric field inside the conductor and the equivalent surface current related to the magnetic field. Existence and uniqueness of solutions, and the convergence of conforming a finite element  boundary element Galerkin discretizations immediately follow. In particular, schemes based on $\curl$conforming edge elements and divergenceconforming surface elements are examined. 
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R. Hiptmair, J. Ostrowski, and R. Quast 
Modelling and Simulation of Induction Heating 
Report No. 149, SFB 382, Universität Tübingen, March 2000 
Keywords : Induction heating, scalar magntic potential, skin effect, boundary elements. 
Abstract: This paper deals with the simulation of induction heating of conducting workpieces with a complex shape and topology. We assume that its conductivity is high, which, owing to the skin effect bars the fields from penetrating deep into the conductors. This justifies the use of a simple magnetostatic model that describes the magnetic field outside the workpiece and the inductor. Field concentrating plates can also be taken into account. After possible holes in the conductors have been patched with cutting surfaces, a scalar magnetic potential can be employed. The actual computation relies on a boundary integral equation of the second kind, discretized by means of piecewise constant boundary elements. Thus we get an approximation of the surface currents. Then we use the skin effect formula to determine the rate of heat generation inside the workpiece. 
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V. Gradinaru and R. Hiptmair 
Whitney Forms on Sparse Grids 
Report No. 153, SFB 382, Universität Tübingen, March 2000. To appear in Numer. Math. 
Keywords : Whitney forms, edge elements, sparse grids 
Abstract: Discrete differential forms provide a natural and canonical approach to the discretization of many physical quantities. If the solution is sufficiently smooth, sparse grid finite elements techniques lead to an improved ratio of storage requirements versus the accuracy of discrete approximations. Interpolation estimates are proved in the context of Whitney forms. The results for Lagrangian finite elements emerge as particular case. We address the influence of quadrature rules used for the evaluation of degrees of freedom. 
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R. Hiptmair 
Multigrid for Eddy Current Computation 
Report No. 154, SFB 382, Universität Tübingen, March 2000. Submitted to Math. Comp. 
Keywords : Eddy current problem, edge elements, multilevel methods, multigrid in H(curl), Helmholtzdecomposition 
Abstract: In [R. Hiptmair, Multigrid method for Maxwell's equations, SIAM J. Numer. Anal., 36 (1999), pp. 204225] a novel multigrid method for discrete H(curl)elliptic boundary value problems has been proposed. These frequently occur in computational electromagnetism, in particular in the context of eddy current simulation. This paper focuses on the analysis of those nodal multilevel decompositions of the spaces of edge finite elements that form the foundation of the multigrid method. This paper provides a significant extension of the existing theory. In particular, asymptotically uniform convergence of the multigrid method with respect to the number of refinement levels can be established in a realistic setting for eddy current problems that features truly nonconducting regions, reentrant corners and complex topologies of the conductor. This is made possible by using approximate Helmholtzdecompositions of the function space H(curl) into an H^{1}regular subspace and gradients. The main results of standard multilevel theory for H^{1}elliptic can then be applied to both subspaces. This yields preliminary decompositions still beyond the finite element setting. Judicious alterations can cure this. 
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R. Hiptmair 
Higher Order Whitney Forms 
Progress in Electromagnetics Research, PIER 32, 271299, 2001 Abstract in Journal of Electromagnetic Waves and Applications, 15(3), 341342, 2001 Also: Report No. 156, SFB 382, Universität Tübingen, August 2000 
Keywords : Discrete differential forms, Whitneyforms, finite elements, edge elements hierarchical basis, pversion 
Abstract: The calculus of differential forms can be used to devise a unified description of discrete differential forms of any order and polynomial degree on simplicial meshes in any spatial dimension. A general formula for suitable degrees of freedom is also available. Fundamental properties of nodal interpolation can be established easily. It turns out that higher order spaces, including variants with locally varying polynomial order, emerge from the usual Whitneyforms by local augmentation. This paves the way for an adaptive pversion approach to discrete differential forms. 
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R. Hiptmair 
Discrete HodgeOperators: An Algebraic Perspective 
Progress In Electromagnetics Research, PIER, 32, 247269, 2001 Abstract in Journal of Electromagnetic Waves and Applications, 15(3), 343344, 2001 
Keywords : Discrete differential forms, discrete vector analysis, finite elements, finite volume methods, error estimates 
Abstract: Discrete differential forms should be used to deal with the discretization of boundary value problems that can be stated in the calculus of differential forms. This approach preserves the topological features of the equations. Yet, the discrete counterparts of the metricdependent constitutive laws remain elusive. I introduce a few purely algebraic constraints that matrices associated with discrete material laws have to satisfy. It turns out that most finite element and finite volume schemes comply with these requirements. Thus convergence analysis can be conducted in a unified setting. This discloses basic sufficient conditions that discrete material laws have to meet in order to ensure convergence in the relevant energy norms. 
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R. Hiptmair 
Multilevel Method for Mixed Eigenproblems 
Report 159, Sonderforschungsbereich 382, January 2001 Submitted to SIAM J. Scientific Computing 
Keywords : Mixed eigenvalue problems, edge elements, RaviartThomas elements, mixed finite elements, preconditioned inverse iteration, multigrid methods 
Abstract: For a Lipschitzpolyhedron $\Omega\subset\mathbb{R}^3$ we consider eigenvalue problems curl\alphacurlu =\lambda\Vu$ and grad\alpha divu = \lambdau$, $\lambda>0$, set in H(curl) and H(div). They are = discretized by means of the conforming finite elements introduced by Nedelec. The preconditioned inverse iteration in its subspace variant is adapted to these problems. A standard multigrid scheme serves as preconditioner. The main challenge arises from the large kernels of the operators curl and div. However, thanks to the choice of finite element spaces these kernels have a direct representation through the gradients/rotations of discrete potentials. This makes it possible to use a multigrid iteration in potential space to obtain approximate projections onto the orthogonal complements of the kernels. There is ample evidence that this will lead to an asymptotically optimal method. Numerical experiments confirm the excellent performance of the method even on very fine grids. 
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R. Hiptmair 
Generators of H_{1} for Triangulated Surfaces: Construction and Classification 
SIAM J. Computing 35 (5), pp. 14051423, 2002. 
Keywords : Cellular homology, nonbounding cycles, linking numbers, surface stream functions, computational electromagnetism 
Abstract:
We consider a bounded Lipschitzpolyhedron O of general
topology equipped with a tetrahedral triangulation that induces a mesh
of the surface S. We seek a maximal set of surface edge cycles that
are independent in H_{1}(S) and bounding with respect to the
exterior of O. We present an algorithm for constructing suitable 1cycles: First, representatives of a basis of the homology group H_{1}(S) are constructed, merely using the combinatorial description of the surface mesh. Then, a duality pairing based on linking numbers is used to determine those combinations that are bounding w.r.t. the exterior$. This is the key to circumventing a triangulation of the exterior region in the computations. For shaperegular, quasiuniform families of meshes, the asymptotic complexity of the algorithm is shown to be O(N^2), where N is the number of edges of the surface mesh. The scheme provides an essential preprocessing step for novel boundary element methods in computational electromagnetism, which rely on discrete divergencefree vectorfields and their description through stream functions. 
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R. Hiptmair and C. Schwab 
Natural BEM for the Electric Field Integral Equation on polyhedra 
Report 200104, Seminar für Angewandte Mathematik, ETH Zürich, April 2001. Submitted to SIAM J. Numer. Anal. 
Keywords : Electric field integral equation, Rumsey's principle, RaviartThomas elements, Hodge decomposition, discrete coercivity 
Abstract: We consider the electric field integral equation on the surface of polyhedral domains and its Galerkindiscretization by means of divergenceconforming boundary elements. With respect to a Hodgedecomposition the continuous variational problem is shown to be coercive. However, this does not immediately carry over to the discrete setting, as discrete Hodge decompositions fail to possess essential regularity properties. Introducing an intermediate semidiscrete Hodge decomposition we can bridge the gap and come up with asymptotic apriori error estimates. Hitherto, those had been elusive for nonsmooth boundaries. 
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A. Buffa (Pavia), R. Hiptmair, T. von Petersdorff (Maryland) and C. Schwab (ETH Zürich) 
Boundary Element Methods for Maxwell Equations in Lipschitz Domains 
Report 200105, Seminar für Angewandte Mathematik, ETH Zürich, July 2001,
submitted to Numer. Math. 
Keywords : Electromagnetic scattering, coercive variational problems Hodge decomposition, Calder\'on projector, RT/BDM surface elements 
Abstract:
We consider the Maxwell equations in a domain with Lipschitz boundary and the
boundary integral oprator A occurring in the Calderon projector. We prove
an infsup condition for A using a Hodge decomposition for tangent fields. We apply this to two types of boundary value problems: the exterior scattering problem by a perfectly conducting body, and the dielectric problem with two different materials in the interior and exterior domain. In both cases we obtain an equivalent boundary equation which has a unique solution. We then consider Galerkin discretizations based on RaviartThomas spaces. We show that these spaces have a discrete Hodge decomposition which are in some sense close to the continuous Hodge decomposition. This property allows us to prove quasioptimal convergence of the resulting boundary element methods. 
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R. Hiptmair 
Coupling of Finite Elements and Boundary Elements in Electromagnetic Scattering 
Report 164, Sonderforschungsbereich 382, July 2001 
Keywords : Electromagnetic scattering, Helmholtz decomposition, Hodge decomposition, Calder\'on projector, symmetric coupling, edge elements, discrete coercivity 
Abstract: We consider the scattering of electromagnetic waves at a dielectric object with a rough surface. We investigate the coupling of a weak formulation of Maxwell's equations inside the scatterer with boundary integral equations that arise from the homogeneous problem in the unbounded region outside the scatterer. The symmetric coupling approach based on the full Calderon projector for Maxwell's equations is employed. By splitting both the electric field inside the scatterer and the surface currents into components of predominantly electric and magnetic nature, we can establish coercivity of the coupled variational problem. Discretization relies on both curlconforming edge elements inside the scatterer and divconforming boundary elements for the surface currents. The splitting idea adjusted to the discrete setting permits us to show uniform stability of the discretized problem. We exploit it to come up with apriori convergence estimates. 
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R. Hiptmair 
Boundary Element Methods for Eddy Current Computation 
in C. Carstensen and S. Funken and W. Hackbusch and R.H.W. Hoppe and P. Monk, eds.,
Cmputational Electromagnetics, vol 28 of Springer Lecture Notes
in Computational Science and Engineering, 2003, pp. 103126 also Report 166, Sonderforschungsbereich 382, August 2001, 
Keywords : Eddy current problem, transmission problems, primal and dual formulations, Seifert surfaces, boundary elements 
Abstract: This paper studies numerical methods for eddy current problems in the case of homogeneous, isotropic, and linear materials. It provides a survey of approaches that entirely rely on boundary integral equations and their conforming Galerkin discretization. The pivotal role of potentials is discussed, as well as the topological issues raised by their use. Direct boundary integral equations and the socalled symmetric coupling of the integral equations corresponding to the conductor and the nonconducting regions is employed. It gives rise to coupled variational problems that are elliptic in suitable trace spaces. This implies quasioptimal convergence of Galerkin boundary element schemes. 
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R. Hiptmair 
Finite elements in computational electromagnetism 
Acta Numerica 2002, pp. 237339 
Keywords : Discrete differential forms, edge elements, Maxwell eigenvalue problem, Maxwell source problem, Weighted regularization 
Abstract: This article discusses finite element Galerkin schemes for a number of linear model problems in electromagnetism. The finite element schemes are introduced as discrete differential forms, matching the coordinate independent statement of Maxwell's equations in the calculus of differential forms. The asymptotic convergence of discrete solutions is investigated theoretically. As discrete differential forms represent a genuine generalisation of conventional Lagrangian finite elements, the analysis is based upon a judicious adaptation of established techniques in the theory of finite elements. Risks and difficulties haunting finite element schemes that do not fit the framework of discrete differential forms are highlighted. 
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R. Hiptmair, J. Metzger 
Automated local mode analysis 
Report 174, Sonderforschungsbereich 382, February 2002 
Keywords : Local mode analysis, grid functions, stencils, discrete symbols, object oriented design 
Abstract: Local mode analysis is an established technique to get quantitative estimates for the convergence rates of multigrid for boundary value problems discretized on regular grids. However, when the discrete differential operators are more complicated than a fivepoint stencil in two dimensions, the computations turn out to be extremely tedious. They are only feasible with the aid of computer software. The foundation for such a code is laid by the theory and implementation presented in this paper: We describe a code that can compute multigrid convergence rates for translation invariant local operators on infinite tensor product grids, the usual setting of local mode analysis. The user only has to specify operator stencils. 
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R. Hiptmair 
Abenteuer Anwendung 
DMVNachrichten, No. 4, 2002, pp. 1924 
Keywords : Graphentherie, diskrete Topologie, Triangulierungen 
Abstract: Dies ist die Geschichte eines Numerikers der sich einer konkreten ingenieurtechnischen Problemstellung annahm. Er wusste Bescheid über Finite Elemente und Randelemente, doch das Problem dachte nicht daran, sich mit diesem Instrumentarium allein lösen zu lassen. Er hätte sagen können, dafür sei er nicht zu zuständig. Doch nachdem dies die Ingenieure nur in ihren Vorurteilen gegen Mathematiker bestärkt hätte, hatte er diese Option nicht. So wagte er sich denn auf das Gebiet der algorithmische Topologie ... 
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R. Hiptmair 
Construction and Classification of Surface Cycles 
Proceedings of the European Symposium on Numerical Methods in
Electromagnetics, Toulouse, France, March 68, 2002, B. Michelsen and
F. Decvele, eds., ONERA, Toulouse, pp. 239244

Keywords : Cellular homology, nonbounding cycles, linking numbers, surface stream functions, computational electromagnetism 
Abstract: See related article 
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R. Hiptmair 
Transmission Problems in Eddy Current Computation 
Proceedings of the European Symposium on Numerical Methods in
Electromagnetics, Toulouse, France, March 68, 2002, B. Michelsen and
F. Decvele, eds., ONERA, Toulouse, pp. 1318.

Keywords : Eddy current problems, primal and dual variational formulations, moving conductors, 
Abstract: In eddy current problems there is a natural distinction between conducting and nonconducting regions, in which the equations of the mathematical model have distinct features. Thus, the issue of coupling discrete models in both regions arises. It its particularly urgent, when different frames of reference are used, as in the case of moving conductors and a Lagrangian perspective. We focus on finite element schemes (FEM) and study primal and dual variational formulations. It turns out that the coupling of primal and dual formulations is straightforward even in the case of nonmatching meshes across the boundaries of conductors, because the coupling is completely taken into account in weak form. This makes primaldual coupling an attractive option for dealing with moving conductors in the context of FEM. 
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R. Hiptmair 
From E to Edge Elements 
Berichte des IZWR, 1, 2003, pp. 3947.

Keywords : Maxwell's equations, Discrete differential forms, Whitney forms 
Abstract: The physical nature of electromagnetic fields suggests differential forms as the natural tool for their mathematical modeling. Cochains on cellular complexes offer a discretization that preserves fundamental topological features of the laws of electromagnetism. However, the constitutive equations cannot be dealt with in this framework, which entails extending cochains to discrete differential forms, whose lowest order variant is known as Whitneyforms. 
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V. Gradinaru and R. Hiptmair 
Multigrid for discrete differential forms on sparse grids 
Computing 71 (2003), pp. 1742 (Also: report Nr. 183, SFB 382, Universität Tübingen, December 2002). 
Keywords : 
Abstract: Discrete differential forms are a generalization of the common H^{1}conforming Lagrangian elements. For the latter Galerkin schemes based on sparse grids are well known, and so are fast iterative multilevel solvers for the discrete Galerkin equations. We extend both the sparse grid idea and the design of multilevel methods to arbitrary discrete differential forms. The focus of this presentation will be on issues of efficient implementation and numerical studies of convergence of multigrid solvers. 
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A. Buffa and R. Hiptmair 
A coercive field integral equation for electromagnetic scattering 
SIAM J. Numer. Anal. 40(2), 621640, 2004. Also: Preprint NI03003, Isaac Newton Institute for Mathematical Sciences, Cambridge, UK, February 2003. 
Keywords : Electromagnetic scattering, combined field integral equations (CFIE), coercivity, boundary element methods, Galerkin scheme 
Abstract: Many boundary integral equation methods used in the simulation of direct electromagnetic scattering of a timeharmonic wave at a perfectly conducting obstacle break down, when applied at frequencies close to a resonant frequency of the obstacle. A remedy is offered by special indirect boundary element methods based on the socalled combined field integral equation. However, hitherto no theoretical results about the convergence of discretized combined field integral equations have been available. In this paper we propose a new combined field integral equation, convert it into variational form, establish its coercivity in the natural trace spaces for electromagnetic fields, and conclude existence and uniqueness of solutions for any frequency. Moreover, a conforming Galerkin discretization of the variational equations by means of $\bDiv$conforming boundary elements can be shown to be asymptotically quasioptimal. This permits us to derive quantitative convergence rates on sufficiently fine, uniformly shaperegular sequences of surface triangulations. 
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A. AlonsoRodriguez, R. Hiptmair, and A. Valli 
Mixed Finite Element Approximation of Eddy Current Problems 
IMA J. Numer. Anal 24(2), 2004. Also: Report 8/2003, Universita degli studi di Milano, Dipartimento di Matematica, March 2003 
Keywords : current problems; mixed finite elements; Langrange multipliers 
Abstract: Finite element approximations of eddy current problems that are entirely based on the magnetic field H are haunted by the need to enforce the algebraic constraint curl H=0 in nonconducting regions. As an alternative to techniques employing combinatorial Seifert (cutting) surfaces in order to introduce a scalar magnetic potential, we propose mixed multifield formulations, which enforce the constraint in the variational formulation. In light of the fact that the computation of cutting surfaces is expensive, the mixed finite element approximation is a viable option despite the increased number of unknowns. 
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R. Hiptmair 
Coupled Boundary Element Scheme for Eddy Current Computation 
Proceedings 2. Kolloquium Elektromagnetische Umformung, Dortmund 28.05.2003

Keywords : boundary element methods, eddy current problem 
Abstract: This article presents the mathematical foundation of a symmetric boundary element method for the computation of eddy currents in a linear homogeneous conductor which is exposed to an alternating magnetic field. Starting from the Abased variational formulation of the eddy current equations and a related transmission problem, the problem inside and outside the conductors can be reformulated as integral equations on the boundary of the conductors. Surface currents occur as new unknowns of this direct formulation. The integral equations can be coupled in a symmetric fashion using the transmission conditions for A and H. The resulting variational problem is elliptic in suitable trace spaces. A conforming Galerkin boundary element discretization is employed, which relies on surface edge elements and provides quasioptimal discrete approximations for the tangential traces of $A and H. Surface stream functions supplemented with cohomology vector fields ensure the vital zero divergence of the discrete equivalent surface currents. Simple expressions allow the computation of approximate total Ohmic losses and surface forces from the discrete boundary data. 
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R. Hiptmair and A. Schädle 
Nonreflecting boundary conditions for Maxwell's equations 
Computing 71 (2003), pp. 165292.

Keywords : Yee scheme, absorbing boundary conditions, fast convolution quadrature 
Abstract: A new discrete nonreflecting boundary condition for the timedependent Maxwell equations describing the propagation of an electromagnetic wave in an infinite homogenous lossless rectangular waveguide with perfectly conducting walls is presented. It is derived from a virtual spatial finite difference discretization of the problem on the unbounded domain. Fourier transforms are used to decouple transversal modes. A judicious combination of edge based nodal values permits us to recover a simple structure in Laplace domain. Using this, it is possible to approximate the convolution in time by a similar fast convolution algorithm as for the standard wave equation. 
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R. Hiptmair and O. Sterz 
Current and Voltage Excitation for the Eddy Current Model 
Int. J. Numer. Modelling 18(1), pp. 121, 2005. Also: Research Report 200307, SAM, ETH Zürich, July 2003 
Keywords : Eddy current model, coupling of circuits and field equations, voltage and current excitation, variational formulations 
Abstract: This paper describes a novel quadrilateral edge element discretisation of Maxwell's equations in which the effects of dispersion are minimised. A modified edge finite element stencil is proposed and it is subsequently shown how this can be expressed in terms of new material coefficients thus allowing us to incorporate both Dirchlet and Neumann boundary conditions in a natural fashion. To demonstrate the success of the proposed procedure, we include a series of numerical examples. First we apply the approach to plane and circular wave propagation problems. Secondly, we apply the approach to a series of electromagnetic scattering problems. For the electromagnetic scattering computations, we monitor the effect of the modified edge finite element stencil on the scattering width output. We use a $hp$edge element code as a benchmark for all our electromagnetic scattering computations. 
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R. Hiptmair and P. Ledger 
A quadrilateral edge element scheme with minimum dispersion 
Research Report 200317, SAM, ETH Zürich

Keywords : Edge elements, generalized finite elements, numerical dispersion, electromagnetic scattering 
Abstract: This paper describes a novel quadrilateral edge element discretisation of Maxwell's equations in which the effects of dispersion are minimised. A modified edge finite element stencil is proposed and it is subsequently shown how this can be expressed in terms of new material coefficients thus allowing us to incorporate both Dirchlet and Neumann boundary conditions in a natural fashion. To demonstrate the success of the proposed procedure, we include a series of numerical examples. First we apply the approach to plane and circular wave propagation problems. Secondly, we apply the approach to a series of electromagnetic scattering problems. For the electromagnetic scattering computations, we monitor the effect of the modified edge finite element stencil on the scattering width output. We use a $hp$edge element code as a benchmark for all our electromagnetic scattering computations. 
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R. Hiptmair and P. Ledger 
Computing of resonant modes for axisymmetric cavities using hpversion finite elements 
Research Report 200315, SAM, ETH Zürich

Keywords : 
Abstract: The computation of the resonant frequencies for open and closed cavities is not a trivial task: Multimaterials and sharp corners all give rise to highly singular eigenfunctions. However, an approach using hpfinite elements is well suited to such problems and, with the correct combination of h and prefinements, it yields the theoretically predicated exponential rates of convergence. In this paper, we present a novel approach to the solution of axisymmetric cavity problems which uses a hierarchic H1 and H(curl) conforming finite element basis. A selection of numerical examples are included and these demonstrate that the exponential rates of convergence are achieved in practice. 
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T. Bubeck, R. Hiptmair and H. Yserentant 
The Finite Mass Mesh Method 
Report 194 Sonderforschungsbereich 256, Universität Tübingen

Keywords : Finite mass method, particle mesh methods, LagrangianEulerian schemes 
Abstract: The finite mass method is a purely Lagrangian scheme for the spatial discretisation of the macroscopic phenomenological laws that govern the flow of compressible fluids. In this article we investigate how to take into account long range gravitational forces in the framework of the finite mass method. This is achieved by incorporating an extra discrete potential energy of the gravitational field into the Lagrangian that underlies the finite mass method. The discretisation of the potential is done in an Eulerian fashion and employs an adaptive tensor product mesh fixed in space, hence the name finite mass mesh method for the new scheme. The transfer of information between the mass packets of the finite mass method and the discrete potential equation relies on numerical quadrature, for which different strategies will be proposed. The performance of the extended finite mass method for the simulation of twodimensional gas pillars under selfgravity will be reported. 
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A. AlonsoRodriguez, R. Hiptmair, and A. Valli 
Hybrid Formulations of Eddy Current Problems 
Report UTM 663 Dipartimento di Matematica, Universita di Trento

Keywords : Timeharmonic eddy current problems, hybrid formulation, mixed finite element approximation. 
Abstract: In this paper we examine the wellknown magnetoquasistatic eddy current model for the behaviour of lowfrequency electromagnetic fields. We restrict ourselves to formulations in the frequency domain and linear materials, but admit rather general topological arrangements. The generic eddy current model allows two dual formulations, which may be dubbed Ebased and Hbased. We investigate socalled hybrid approaches that combine both formulations by means of coupling conditions across the boundaries of conducting regions. The resulting continuous and discrete variational formulations will be discussed. Particular emphasis is laid on difficulties arising from the topology of the conducting regions. 
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A. Buffa and R. Hiptmair 
Coercive combined field integral equations 
Research report 200308, SAM, ETH Zürich. Submitted to Numer. Math. 
Keywords : Acoustic scattering, indirect boundary integral equations, combined field integral equations (CFIE), coercivity, boundary element methods, Galerkin schemes. 
Abstract: Many boundary integral equations for exterior Dirichlet and Neumann boundary value problems for the Helmholtz equation suffer from a notorious instability for wave numbers related to interior resonances. The socalled combined field integral equations are not affected. However, if the boundary $\Gamma$ is not smooth, the traditional combined field integral equations for the exterior Dirichlet problem do not give rise to an $\xLzwei{\Gamma}$coercive variational formulation. This foils attempts to establish asymptotic quasioptimality of discrete solutions obtained through conforming Galerkin boundary element schemes. This article presents new combined field integral equations on twodimensional closed surfaces that possess coercivity in canonical trace spaces. The main idea is to use suitable regularizing operators in the framework of both direct and indirect methods. This permits us to apply the classical convergence theory of conforming Galerkin methods. 
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R. Hiptmair 
Coercive combined field integral equations 
J. Numer. Math. 11(2), 2003, pp. 115134

Keywords : Acoustic scattering, indirect boundary integral equations, combined field integral equations (CFIE), coercivity, boundary element methods, Galerkin schemes. 
Abstract: Many boundary integral equations for exterior Dirichlet and Neumann boundary value problems for the Helmholtz equation suffer from a motorious instability for wave numbers related to interior resonances. The socalled combined field integral equations are not affected. This article presents combined field integral equations on twodimensional closed surfaces that possess coercivity in canonical trace spaces. For the exterior Dirichlet problem the main idea is to use suitable regularizing operators in the framework of an indirect method. This permits us to apply the classical convergence theory of conforming Galerkin methods. 
Link 
B. CranganuCretu and R. Hiptmair 
Direct Boundary Integral Equation Method for Electromagnetic Scattering at Partly Coaeted Dielectric Objects 
Reaearch report 200406, SAM, ETHZ Submitted to Computing and Visualization in Science 
Keywords : Electromagnetic scattering, direct boundary integral equations, Galerkin boundary element method (BEM) 
Abstract: We present a new variational direct boundary integral equation approach for solving the scattering and transmission problem for dielectric objects partially coated with a PEC layer. The main idea is to use to use the electromagnetic Calderon projector along with transmission conditions for the electromagnetic fields. This leads to a symmetric variational formulation which lends itself to Galerkin discretization by means of divergenceconforming discrete surface currents. A wide array of numerical experiments confirms the efficacy of the new method. 
Link 
R. Hiptmair and J. Ostrowski 
Coupled Boundary Element Scheme for Eddy Current Computation 
To appear in J. Engr. Math. Extended abstract in proceedings of Fourth UK Conference on Boundary Integral Methods, Salford, UK, Sep 1516, 2003 
Keywords : Eddy current problem, boundary elements, divergenceconforming boundary elements, surface stream functions, homology spaces 
Abstract: The mathematical foundation of a symmetric boundary element method for the computation of eddy currents in a linear homogeneous conductor which is exposed to an alternating magnetic field is presented. Starting from the $\mathbf{A}$based variational formulation of the eddy current equations and a related transmission problem, the problem inside and outside the conductors is reformulated in terms of integral equations on the boundary of the conductors. Surface currents occur as new unknowns of this direct formulation. The integral equations can be coupled in a symmetric fashion using the transmission conditions for the vector potential $\mathbf{A}$ and the magnetic field $\mathbf{H}$. The resulting variational problem is elliptic in suitable trace spaces. A conforming Galerkin boundary element discretization is employed, which relies on surface edge elements and provides quasioptimal discrete approximations for the tangential traces of $\mathbf{A}$ and $\mathbf{H}$. Surface stream functions supplemented with cohomology vector fields ensure the vital zero divergence of the discrete equivalent surface currents. Simple expressions allow the computation of approximate total Ohmic losses and surface forces from the discrete boundary data. 
No link 