A Black-Box Multigrid Preconditioner for the Biharmonic Equation

Abstract

We examine the convergence characteristics of a preconditioned Krylov subspace solver applied to the linear systems arising from low-order mixed finite element approximation of the biharmonic problem. The key feature of our approach is that the preconditioning can be realized using any “black-box” multigrid solver designed for the discrete Dirichlet Laplacian operator. This leads to preconditioned systems having an eigenvalue distribution consisting of a tightly clustered set together with a small number of outliers. Numerical results show that the performance of the methodology is competitive with that of specialized fast iteration methods that have been developed in the context of biharmonic problems.