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Local projection methods on layer-adapted meshes for higher order discretisations of convection-diffusion problems. (English) Zbl 1188.65154

This paper studies singularly perturbed boundary value problems in the unit square discretized with higher-order finite elements. The solution of the considered convection-diffusion equation exhibits exponential layers in the vicinity of the boundary. A local projection method is used for stabilizing the discretized problem. Two types of layer adapted meshes, the Shishkin mesh, which is piecewise uniform and adapted to the boundary layers, and the Bakhvaloy-Shishkin mesh, which is uniform only on the course part and graded towards the boundary in the fine parts of the mesh, are examined.
A proof for the error decay rate of the solution of the stabilized discretization and a special interpolation of the solution of the continuous problem is given. Numerical results in two dimensions for a problem for which the exact solution is available illustrate the properties of the method for both types of meshes and support the theoretically shown convergence orders.

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs

Software:

UMFPACK; MooNMD
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Full Text: DOI

References:

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