Local projection methods on layer-adapted meshes for higher order discretisations of convection--diffusion problems

  • Authors:

  • Gunar Matthies

  • Affiliations:

  • Universität Kassel, Fachbereich 17 Mathematik, Heinrich-Plett-Straße 40, 34132 Kassel, Germany

  • Venue:
  • Applied Numerical Mathematics
  • Year:
  • 2009

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Abstract

We consider singularly perturbed convection-diffusion problems in the unit square where the solutions show the typical exponential layers. Layer-adapted meshes (Shishkin and Bakhvalov-Shishkin meshes) and the local projection method are used to stabilise the discretised problem. Using enriched Q"r-elements on the coarse part of the mesh and standard Q"r-elements on the remaining parts of the mesh, we show that the difference between the solution of the stabilised discrete problem and a special interpolant of the solution of the continuous problem convergences @e-uniformly with order O(N^-^(^r^+^1^/^2^)) on Bakhvalov-Shishkin meshes and with order O(N^-^(^r^+^1^/^2^)+N^-^(^r^+^1^)ln^r^+^3^/^2N) on Shishkin meshes. Furthermore, an @e-uniform convergence in the @e-weighted H^1-norm with order O((N^-^1lnN)^-^r) on Shishkin meshes and with order O(N^-^r) on Bakhvalov-Shishkin meshes will be proved. Numerical results which support the theory will be presented.