Adaptive Bem-based Fem on Polygonal Meshes from Virtual Element Methods

Polygonal meshes are especially suited for the discretization of boundary value problems in adaptive mesh refinement strategies. Such meshes are very flexible and incorporate hanging nodes naturally. But only a few approaches are available that handle polygonal discretizations in this context. The BEM-based Finite Element Method (FEM) and a residual based error estimate are reviewed in the presentation. This a posteriori error estimate is reliable and efficient on polygonal meshes and can be applied in adaptive FEM strategies. Furthermore, the BEM-based FEM is applicable on such general meshes and gains its flexibility by implicitly defined trial functions. They are given as solutions of local Dirichlet problems related to the global differential operator. These local problems are treated by means of Boundary Element Methods (BEM) in the realization. In the numerical experiments the test problems of the recent publication on adaptive Virtual Element Methods by L. Beirão da Veiga and G. Manzini [ESAIM Math. Model. Numer. Anal., 49(2):577–599, 2015] are considered in an adaptive BEM-based FEM simulation. The experiments show optimal rates of convergence for uniform and adaptive mesh refinement, where the latter one yields, in particular, very local mesh adaptation.