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Multigrid algorithm for the solution of partial differential equations on complicated domains:
| The research project is devoted to the development and analysis of a new multigrid method for solving systems of linear equations as they arise from discretisations of boundary value problems on complicated domains. The finite element method is commonly employed to discretise these equations due to its flexibility of resolving complicated geometries. The dimension of the resulting system of discrete equations is huge. These equations should be solved by using a multigrid method. The efficiency of multigrid methods is based on a multiscale discretisation of the problem.
| | However, in many practical applications coarsescale discretisations are not available. Therefore, so-called composite finite elements have been introduced. These elements allow coarsescale discretisations of problems with complicated geometries by only a few unknowns, the minimal number being independent of the number and size of geometric details.
| | During this project, the convergence of multigrid methods with composite finite elements should be investigated and generalized to interesting problems as the Stokes equation and the biharmonic equation. |
| Keywords: composite finite elements, multigrid methods, coarsening, algebraic multigrid, approximation property, convergence theory |
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