Robust a posteriori error estimates for conforming discretizations of a singularly perturbed reaction-diffusion problem
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Robust a posteriori error estimates for conforming discretizations of a singularly perturbed reaction-diffusion problem on anisotropic meshes Jikun Zhao · Shaochun Chen
Received: 16 January 2013 / Accepted: 25 September 2013 © Springer Science+Business Media New York 2013
Abstract In this paper, we derive robust a posteriori error estimates for conforming approximations to a singularly perturbed reaction-diffusion problem on anisotropic meshes, since the solution in general exhibits anisotropic features, e.g., strong boundary or interior layers. Based on the anisotropy of the mesh elements, we improve the a posteriori error estimates developed by Cheddadi et al., which are reliable and efficient on isotropic meshes but fail on anisotropic ones. Without the assumption that the mesh is shape-regular, the resulting mesh-dependent error estimator is shown to be reliable, efficient and robust with respect to the reaction coefficient, as long as the anisotropic mesh sufficiently reflects the anisotropy of the solution. We present our results in the framework of the vertex-centered finite volume method but their nature is general for any conforming one, like the piecewise linear finite element one. Our estimates are based on the usual H(div)-conforming, locally conservative flux reconstruction in the lowest-order Raviart-Thomas space on a dual mesh associated with the original anisotropic simplex one. Numerical experiments in 2D confirm that our estimates are reliable, efficient and robust on anisotropic meshes. Keywords Robust a posteriori error estimates · Anisotropic meshes · Vertex-centered finite volume method · Singularly perturbed reaction-diffusion problem · Conforming discretization Mathematics Subject Classifications (2010) 65N15 · 65N30 Communicated by: Tomas Sauer This work was supported by National Natural Science Foundation of China (11371331) J. Zhao () · S. Chen School of Mathematics and Statistics, Zhengzhou University, Zhengzhou 450001, China e-mail: [email protected] S. Chen e-mail: [email protected]
J. Zhao, S. Chen
1 Introduction In this paper, we consider the reaction-diffusion problem with homogeneous Dirichlet boundary condition −u + r 2 u = f, in , (1.1) u = 0, on ∂, where ⊂ Rd , (d = 2, 3), is a polygonal (polyhedral) domain (open, bounded, and connected set), r ∈ L∞ () a reaction coefficient (r ≥ 0), and f ∈ L2 () a source term. We denote by rmin,S and rmax,S the best positive constants such that rmin,S ≤ r ≤ rmax,S on a given subdomain S of . If r 1, then problem Eq. 1.1 becomes a singularly perturbed problem and the solution u is anisotropic, i.e., it exhibits boundary or interior layers. Then, in order to reflect the anisotropy of the solution, a proper anisotropic discretization are particularly useful or even mandatory, in the context of the finite element method, which leads to anisotropic meshes [4]. The purpose of this paper is to derive a posteriori error estimates for the vertexcentered finite volume approximation of problem Eq. 1.1 on anisotropic meshes, with exten
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