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On a posteriori error estimators in the infinite element method on anisotropic meshes. (English) Zbl 0934.65122

Within the context of the Poisson equation on a bounded domain in \(\mathbb{R}^2\), the authors consider an anisotropic finite element triangulation. One of the conditions of the triangulation ensures that the anisotropic mesh does not change too much. Standard finite element spaces of continuous and piecewise linear or quadratic shape functions are used. The crucial point about anisotropic meshes is that all classical estimators deteriorate as the aspect ratio of elements becomes unbounded. The authors aim to derive a posteriori estimates for the error \(e = u - u_h\) in terms of local indicators \(\eta_\Lambda\) satisfying \[ m_1\|De\|^2 - T(h,f) \leq \sum_{\Lambda} \eta_\Lambda^2 \leq m_2\|De\|^2 + T(h,f), \tag{1} \] in which \(\Lambda\) denotes an element in the triangulation, and \(T\) is a function depending on the mesh size and the data.
It is shown first that the constants in the estimate (1) depend on aspect ratio \(a\) for residual-based estimates. The authors then proceed to introduce a nonlocal error estimator, based on the third indicator of R. E. Bank and A. Wieser [Math. Comput. 44, 283-301 (1985; Zbl 0569.65079)], which can be computed economically on both isotropic and anisotropic meshes. On anisotropic meshes the estimator shows a significant propagation of local errors along the direction orthogonal to the longer sides of the triangles. This shows that local a posteriori estimation is not possible when the standard Dirichlet norm is used. The authors derive an estimate which confirms the reliability and efficiency of the nonlocal indicator.
The paper concludes with a numerical example which confirms the theoretical result.

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs

Citations:

Zbl 0569.65079
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