Developments and challenges in BEM-based Finite Element Methods on general meshes

Developments and challenges in BEM-based Finite Element Methods on general meshes

In the development of numerical methods to solve boundary value problems the requirement of flexible mesh handling gains more and more importance. The BEM-based finite element method is one of the new promising strategies which yields conforming approximations on polygonal and polyhedral meshes, respectively. This flexibility is obtained by special trial functions which are defined implicitly as solutions of local boundary value problems related to the underlying differential equation. The first part of the presentation gives a short introduction into the BEM-based FEM and deals with recent developments. Here, the definitions of lower as well as higher order trial functions are discussed, and the advantages in an adaptive strategy are shown using a suitable residual error estimator. In the second part, the focus lies on challenges given by ongoing work. This includes H(div)-conforming approximations for a mixed formulation of BEM-based FEM and a new generalization to three space dimensions. This generalization makes use of a stepwise construction of trial functions and promises advantageous properties for convection dominated diffusion, for example.