20160407
In the development of numerical methods to solve boundary value problems the requirement of flexible mesh handling gains more and more importance. The BEMbased finite element method [1] is one of the new promising strategies which yield 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 in the spirit of Trefftz [2]. Due to this construction, the approximation space already inherit some properties of the unknown solution. These implicitly defined trial functions are treated by means of boundary element methods (BEM) in the realisation.
The presentation gives a short introduction into the BEMbased FEM and deals with recent challenges and developments. The basic idea in the construction of trial functions is generalised, and thus, trial functions of arbitrary order are obtained [3]. Furthermore, by using a posteriori error estimates, it is possible to achieve optimal rates of convergence even for problems with nonsmooth solutions on adaptive refined polygonal meshes [4]. Several numerical examples confirm the theoretical results.
References
[1] S. Rjasanow and S. Weißer. FEM with Trefftz trial functions on polyhedral elements. Journal of Computational and Applied Mathematics, 263:202217, 2014.
[2] E. Trefftz. Ein Gegenstück zum Ritzschen Verfahren. In Proceedings of the 2nd International Congress of Technical Mechanics, Orell Fussli Verlag, 131137, 1926.
[3] S. Rjasanow and S. Weißer. Higher order BEMbased FEM on polygonal meshes.SIAM Journal on Numerical Analysis, 50(5):23572378, 2012.
[4] S. Weißer. Residual error estimate for BEMbased FEM on polygonal meshes. Numer. Math., 118(4):765788, 2011.
Acknowledgement
This talk is based on joint research together with Steffen Weißer.
Category: CE SeminarTechnische Universität Darmstadt
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