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A posteriori error estimation and adaptive mesh-refinement techniques. (English) Zbl 0811.65089

To make the ideas more transparent the author restricts the analysis to a simple model problem: a conforming finite element method for the two- dimensional Poisson equation with mixed Dirichlet-Neumann boundary conditions.
There are three a posteriori error estimators which are based on the evaluation of suitable local residuals. It is proved that up to higher- order terms all of them yield global upper and local lower bounds for the true error and that these estimators are all equivalent.
Using these estimators the adaptive mesh-refinement techniques are constructed, that allow to detect local singularities of the solution and to appropriately refine the grid near these singularities. In the end two examples are given, that prove the efficiency of the error estimators and the mesh-refinement techniques.
Reviewer: V.Makarov (Kiev)

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
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
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