Adaptive BEM-based FEM on general polygonal meshes and a residual error estimator

Adaptive BEM-based FEM on general polygonal meshes and a residual error estimator

We briefly introduce a special finite element method that solves the stationary isotropic heat equation with Dirichlet boundary conditions on arbitrary polygonal and polyhedral meshes. The method uses a space of locally harmonic ansatz functions to approximate the solution of the boundary value problem. These ansatz functions are constructed by means of boundary integral formulations. Due to this choice, the proposed finite element method can be used on general polygonal non-conform meshes. Hanging nodes are treated quite naturally and the material properties are assumed to be constant on each element.
In a second step we focus on uniform and adaptive mesh refinement. One important point is the treatment of these arbitrary elements. We propose a method to refine polygonal bounded elements which are convex.
In order to do adaptive mesh refinement it is essential to look at a posteriori error estimates. Standard methods are based on triangular or quadrilateral meshes. The challenging part is to handle the arbitrary polygonal and polyhedral meshes. We generalize the ideas of residual error estimators and use them in numerical examples.