On the quality of local flux reconstructions for guaranteed error bounds. (English) Zbl 1363.65193

Vejchodský, Tomáš

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). 242-255 (2015).

The paper considers reaction-diffusion equations discretized by the finite element method and studies the quality of guaranteed complementary error bounds whose values are determined by suitable flux reconstructions. The performance of the local flux reconstruction of M. Ainsworth and T. Vejchodský [“Robust error bounds for finite element approximation of reaction-diffusion problems with non-constant reaction coefficient in arbitrary space dimension”, Comput. Methods Appl. Mech. Eng. 281, 184–199 (2014)] and the reconstruction of D. Braess and J. Schöberl [Math. Comput. 77, No. 262, 651–672 (2008; Zbl 1135.65041)] are compared numerically. The efficiency of both these flux reconstructions is compared with the optimal flux reconstruction, which is computed as the solution of a global minimization problem.
For the entire collection see [Zbl 1329.00187].

Reviewer: Jan Brandts (Amsterdam)

MSC:

65N15	Error bounds for boundary value problems involving PDEs
65N30	Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J25	Boundary value problems for second-order elliptic equations
35B25	Singular perturbations in context of PDEs

Keywords:

a posteriori error estimates; complementarity; index of effectivity; elliptic problem; reaction-diffusion equations; singular perturbation; finite element method

Citations:

Zbl 1135.65041