Abstract
A multigrid method for an elliptic linear boundary value problem is presented. A consistent lowest-order mixed-finite-element discretisation is used on Cartesian locally refined grids in up to three space dimensions. The solution of the indefinite system of equations is reformulated as a problem of constrained minimisation. The constraint is satisfied exactly after one multigrid cycle; the functional is reduced iteratively by smoothing and coarse-grid corrections. By a suitable choice of prolongation and restriction operators, all corrections on coarser levels also reduce the functional within the constraint. This approach leads to a non-standard convergence proof which also holds for the variable-coefficient case. The proof does not predict the actual convergence rate, but shows that the functional will never increase after a multigrid cycle, while the constraint is satisfied exactly after the first multigrid cycle. The conditions required for convergence allow some freedom in choosing the restriction and prolongation operators, which can be used to improve the convergence rate. This leads to operator weighed restriction and prolongation operators in a novel manner. Some numerical examples are presented to demonstrate the robustness of the method.
Published Version
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