Multigrid methods for nonconforming finite elements
Thesis/Dissertation
·
OSTI ID:5903111
Approximate solutions of elliptic boundary value problems can be obtained by using finite elements. In the multigrid method, and algorithm is constructed recursively to solve a sequence of discretized problems that comes from a family of triangulations of the domain of the elliptic boundary value problem. The algorithm has two main features: smoothing on the current grid and coarser-grid error correction. The smoothing step has the effect of damping out the oscillatory part of the error. The smooth part of the error can be accurately captured on the coarser grid. Furthermore, since the error correction is done on a coarser grid, less computational work is involved. The multigrid method is optimal in the sense that the cost of computation is proportional to the number of unknowns. For higher order elliptic boundary value problems, it is preferable to use nonconforming finite elements. The main difficulty in developing a multigrid method for nonconforming finite elements is that the finite element spaces are not nested. Therefore an intergrid transfer operator other than the natural injection is needed to transfer the functions between grids. In this dissertation, three properties of the intergrid transfer operator crucial to the construction of a multigrid theory for noncomforming finite elements are identified. The intergrid transfer operator obtained by local averaging is shown to have these properties for three methods: P1 nonconforming elements and the Fortin-Soulie element for the Laplace operator and the Morley element for the biharmonic operator. Full multigrid convergence and the optimality of these methods are demonstrated.
- Research Organization:
- Michigan Univ., Ann Arbor, MI (USA)
- OSTI ID:
- 5903111
- Country of Publication:
- United States
- Language:
- English
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