A Particle-Partition of Unity Method Part V: Boundary Conditions

M. Griebel and M. A. Schweitzer
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M. Griebel: Universität Bonn, Institut für Angewandte Mathematik
M. A. Schweitzer: Universität Bonn, Institut für Angewandte Mathematik

A chapter in Geometric Analysis and Nonlinear Partial Differential Equations, 2003, pp 519-542 from Springer

Abstract: Summary In this sequel to [12, 13, 14, 15] we focus on the implementation of Dirichlet boundary conditions in our partition of unity method. The treatment of essential boundary conditions with meshfree Galerkin methods is not an easy task due to the non-interpolatory character of the shape functions. Here, the use of an almost forgotten method due to Nitsche from the 1970’s allows us to overcome these problems at virtually no extra computational costs. The method is applicable to general point distributions and leads to positive definite linear systems. The results of our numerical experiments, where we consider discretizations with several million degrees of freedom in two and three dimensions, clearly show that we achieve the optimal convergence rates for regular and singular solutions with the (adaptive) h-version and (augmented) p-version.

Keywords: Convergence Rate; Dirichlet Boundary Condition; Discontinuous Galerkin Method; Meshfree Method; Essential Boundary Condition (search for similar items in EconPapers)
Date: 2003
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DOI: 10.1007/978-3-642-55627-2_27

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