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BEM-based FEM: residual error estimates and extensions to higher order approximation

Friday, September 9, 1PM – 2PM
6.304

Stefan Weisser, Universität des Saarlandes (Germany)

The interest in more flexible and general meshes for the numerical approximation of boundary value problems has been increased. In recent years, several methods have been developed which handle polygonal and polyhedral meshes. In most cases, these are non-conforming methods like Mimetic Finite Diff erence or discontinuous Galerkin methods, see e.g. [1].

In this research, we review the so called BEM-based fi nite element method which is applied to the stationary isotropic heat equation with mixed Dirichlet and Neumann boundary conditions on arbitrary polygonal and polyhedral meshes. The method uses a space of locally harmonic trial functions to approximate the solution of the boundary value problem. These trial functions are constructed by means of boundary integral formulations. Due to this choice, the proposed fi nite element method can be used on general polygonal non-conform meshes. In the numerics, boundary element methods (BEM) are used to build up the local sti ffness matrixes. According to the use of polygonal elements, hanging nodes are treated quite naturally and even improve the approximation.

In order to do adaptive mesh refi nement, it is essential to look at a posteriori error estimates. Standard methods are developed for triangular and quadrilateral meshes. The challenging part is to handle arbitrary polygonal and polyhedral meshes. We generalise the ideas of residual error estimates and use them in numerical examples as presented in [2].

In the current presentation, the BEM-based FEM is a method of rst order. Therefore, we discuss extensions of the special fi nite element method to obtain higher order convergence.

References [1] L. Demkowicz and J. Gopalakrishnan: Analysis of the DPG Method for the Poisson Equation. ICES REPORT 10-37, The Institute for Computational Engineering and Sciences, The University of Texas at Austin, September 2010 [2] S. Weisser: Residual error estimate for BEM-based FEM on polygonal meshes. Numerische Mathematik, 118:765-788, 2011, DOI: 10.1007/s00211-011-0371-6.

Hosted by Leszek Demkowicz