Variational-difference for solving geometrically complex problems on simple grids, with applications to macromolecules and nanoparticles

I. Tsukerman, University of Akron, USA

Wednesday, June 18
at 16.30
in HG E1.2

Numerical solution of boundary value problems can be made substantially more accurate if some features of the exact solution are known a priori and are reflected in the choice of local approximating functions. (The physical examples include point charge singularities in electrostatics, field jumps at material interfaces, boundary layers, and so on.) The motivation (and motif) of this talk is freedom of approximation that is a way to fuse any desired local approximations into a global variational-difference scheme. In particular, instead of representing material interfaces geometrically on conforming meshes, it may be beneficial to approximate these interfaces algebraically by a judicious choice of basis functions.

In summary, the proposed class of schemes combines (i) a domain cover by (generally overlapping) subdomains (patches), as in Generalized FEM; (ii) suitable local approximating functions within each patch (e.g. constructed by a spatial mapping in case of a sharp variation of material parameters); (iii) a general moment method; (iv) nodal values as the main degrees of freedom.

The most interesting practical applications include electrostatic and magnetostatic problems with multiple nanoparticles, and simulation of macromolecules with explicit and implicit solvent models. A few numerical examples will be presented. The intention is to make the talk understandable and hopefully interesting, not only to numerical analysts but also to researchers in various areas of computational chemistry and physics.