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Local problems on stars: A posteriori error estimators, convergence, and performance. (English) Zbl 1019.65083

The authors consider the numerical solution of a generalised two-dimensional Poisson equation with the Dirichlet boundary condition over a bounded polygonal domain by means of triangular finite elements and produce estimates for the error. Estimators with a self-equilibration property are suggested and a convergent algorithm is given for the solution of the problem. A measure of the error is defined and an indication as to how it may be reduced is given, together with rules for oscillation reduction.
Results are established for boundary stars. The paper closes with the results of some numerical experiments and it is suggested that the performance of the error estimators and quasi-optimality of the algorithm is good. The examples of the application of the method, the problem of a crack and a problem involving discontinuous coefficients are used.

MSC:

65N15 Error bounds for boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods 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
65Y20 Complexity and performance of numerical algorithms

Software:

ALBERT
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References:

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