Plane wave discontinuous Galerkin methods

Description

The numerical simulation of the propagation and interaction of acoustic and electromagnetic waves described by the Helmholtz and Maxwell equations using standard finite elements methods is affected by some serious problems (e.g., numerical dispersion, pollution effect) that make it computationally too expensive in many practical cases. Some methods that incorporate properties of the solutions into the discretization have been developed in order to overcome these obstacles. One of these methods is the plane wave discontinuous Galerkin method (PWDG), which uses finite dimensional spaces of plane wave functions within a discontinuous Galerkin framework.

We derived and analyzed the h- and the p-versions of the PWDG for the homogeneous Helmholtz equation in two and three dimensional domains with impedance boundary conditions. The proof of algebraic orders of convergence in h (the meshwidth) and p (the local number of plane waves) follows from best approximation estimates for plane and circular/spherical waves spaces. These estimates are proved using Vekua's theory for the N-dimensional Helmholtz operator, harmonic polynomial approximation results and a careful residual estimates of the Jacobi-Anger expansion.

A natural generalization of the PWDG method can be carried out for the Maxwell equations (and even for time-harmonic elastic wave equations). The error analysis of the method requires new wavenumber-explicit stability and regularity results for the impedance boundary value problem that relies on Rellich-type estimates.

Publications / Preprints or SAM-Reports

• A. Moiola, Plane wave approximation in linear elasticity, SAM Report 2011-20, ETH Zurich.
• R. Hiptmair, A. Moiola, I. Perugia, Error analysis of Trefftz-discontinuous Galerkin methods for the time-harmonic Maxwell equations, SAM Report 2011-09, ETH Zurich.
• R. Hiptmair, A. Moiola, I. Perugia, Stability results for the time-harmonic Maxwell equations with impedance boundary conditions, SAM Report 2010-39, ETH Zurich.
• A. Moiola, R. Hiptmair, I. Perugia, Plane wave approximation of homogeneous Helmholtz solutions, 25PV10/23/0, I.M.A.T.I.-C.N.R. Pavia (2010).
• A. Moiola, R. Hiptmair, I. Perugia, Vekua theory for the Helmholtz operator, 24PV10/22/0, I.M.A.T.I.-C.N.R. Pavia (2010).
• A. Moiola, Approximation properties of plane wave spaces and application to the analysis of the plane wave discontinuous Galerkin method, SAM Report 2009-06, ETH Zurich.

References

• R. Hiptmair, A. Moiola, I. Perugia, Plane wave discontinuous Galerkin methods for the 2D Helmholtz equation: analysis of the p-version, Siam J. Numer. Anal., 49 (2011), pp. 264-284 (link to journal).
• C. Gittelson, R. Hiptmair and I. Perugia, Plane wave discontinuous Galerkin methods: analysis of the h-version, M2AN Math. Model. Numer. Anal. 43 (2009), pp. 297-332 (link to journal).
  

Funding

Partial support by the Swiss National Science Foundation.

Contacts

R. Hiptmair

A. Moiola

I. Perugia