A multi-space error estimation approach for meshfree methods. (English) Zbl 1363.65192

Rüter, Marcus; Chen, Jiun-Shyan

Brandts, J. (ed.) et al., Proceedings of the international conference ‘Applications of mathematics’, Prague, Czech Republic, November 18–21, 2015. In honor of the birthday anniversaries of Ivo Babuška (90), Milan Práger (85), and Emil Vitásek (85). Prague: Czech Academy of Sciences, Institute of Mathematics (ISBN 978-80-85823-65-3). 194-205 (2015).

Summary: Error-controlled adaptive meshfree methods are presented for both global error measures, such as the energy norm, and goal-oriented error measures in terms of quantities of interest. The meshfree method chosen in this paper is the reproducing kernel particle method, since it is based on a Galerkin scheme and therefore allows extensions of quality control approaches as already developed for the finite element method. Our approach of goal-oriented error estimation is based on the well-established technique using an auxiliary dual problem. To keep the formulation general and to add versatility, a multi-space approach is used, where the dual problem is solved numerically using a different approximation space than the one employed in the associated primal problem. This can be realized with meshfree methods at no additional cost. Possible merits of this multi-space approach are discussed and an illustrative numerical example is presented.
For the entire collection see [Zbl 1329.00187].

MSC:

65N15	Error bounds for boundary value problems involving PDEs
65N50	Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs
46E22	Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces)
35J30	Higher-order elliptic equations
65N30	Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs

Keywords:

reproducing kernel particle method; meshfree methods; goal-oriented error estimation; dual problem; Galerkin scheme; finite element method; numerical example