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A priori mesh grading for an elliptic problem with Dirac right-hand side. (English) Zbl 1229.65203

The authors are concerned with the finite element method analysis of the homogeneous Dirichlet problem attached to the Poisson equation with the Dirac distribution in the right hand side. As the finite element method error estimates for this problem are nonstandard and the quasi-uniform meshes are unsuitable, the authors first consider the notion of graded meshes with respect to the point where the Dirac measure is concentrated. Making use of graded meshes error estimates, the authors prove an almost optimal order of convergence of the method. Two numerical examples are carried out in order to underline the capabilities of the method.

MSC:

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
65N15 Error bounds for boundary value problems involving PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
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