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Local a posteriori estimates for pointwise gradient errors in finite elemet methods for elliptic problems. (English) Zbl 1144.65068

The paper is concerned with sharp residual-based a posteriori upper bounds for the local gradient error in finite element methods for elliptic problems. The author considers the boundary value problem for a second-order linear elliptic equation, \(- \mathrm{div} (A \nabla u) = f\), in a bounded domain of \(\mathbb R^2\) or \(\mathbb R^3\) with homogeneous Dirichlet boundary condition. The quantity \(\| \nabla (u-u_h)\| _{L^\infty(D)}\), where \(u_h\) is a conforming finite element approximation of \(u\) and \(D\) a subset of the computational domain, is divided into a computable local residual term and a global pollution term which measures the finite element error in a weaker norm. The residual term is sharply bounded using a mesh-dependent weight. In specific situations the pollution term may also be bounded by computable residual estimators which do not employ precise knowledge of singularities arising at re-entrant corners of the domain. The finite element mesh is only required to be simplicial and shape-regular so that highly graded and unstructured meshes are allowed. Some computational experiments for the Laplace operator are included illustrating the effectiveness of the error estimators for controlling local gradient errors in plane polygonal crack domains.

MSC:

65N15 Error bounds for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations

Software:

ALBERTA; ALBERT
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References:

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