Robust Domain Decomposition Preconditioners for hp Finite Element Approximations on Highly Anisotropic Meshes


Solutions of elliptic boundary value problems in polyhedral domains have corner and edge singularities. Singularities may also arise at material interfaces. In addition, boundary layers often arise in flows with moderate or high Reynolds numbers and in conductor materials with very large conductivity jumps, at faces, edges, corners, and material interfaces. Suitably graded anisotropic meshes, geometrically refined toward corners, edges, faces, and/or interfaces, lead to an exponential rate of convergence of hp finite element approximations. In many practical applications, it is not merely a matter of speeding up convergence, but of making computations possible, since simple h or p approximations on isotropic meshes would require a prohibitively large number of unknowns.

The bottleneck for computations involving such problems is often the solution of the corresponding algebraic systems, which have huge condition numbers due to the simultaneous effect of the large number of unknowns, the large coefficient jumps, the huge aspect ratios of the mesh, and small parameters. Robust preconditioning is mandatory.

In recent years, we have been able to devise a successful robust domain decomposition preconditioning strategy for some hp approximations of scalar problems on highly anisotropic two- and three-dimensional meshes. More recently, this strategy has been extended to some edge element approximations of electromagnetic problems. Thanks to a particular choice of the subdomains and of the coarse solvers, our preconditioners ensure quasioptimality, scalability, robustness with respect to coefficient jumps and huge aspect ratios of the meshes. We illustrate these features through numerical tests.

Top of Page