Clemens Hofreither

Clemens Hofreither, Ulrich Langer, Steffen Weißer

We present a new discretization method for homogeneous convection-diffusion-reaction boundary value problems in 3D that is a non-standard finite element method with PDE-harmonic shape functions on polyhedral elements. The element stiffness matrices are constructed by means of local boundary element techniques. Our method, which we refer to as a BEM-based FEM, can therefore be considered a local Trefftz method with element-wise (locally) PDE-harmonic shape functions. The Dirichlet boundary data for these shape functions is chosen according to a convection-adapted procedure which solves projections of the PDE onto the edges and faces of the elements. This improves the stability of the discretization method for convection-dominated problems both when compared to a standard FEM and to previous BEM-based FEM approaches, as we demonstrate in several numerical experiments.

Clemens Hofreither, Stefan Takacs, Walter Zulehner

We consider geometric multigrid methods for the solution of linear systems arising from isogeometric discretizations of elliptic partial differential equations. For classical finite elements, such methods are well known to be fast solvers showing optimal convergence behavior. However, the naive application of multigrid to the isogeometric case results in significant deterioration of the convergence rates if the spline degree is increased. Recently, a robust approximation error estimate and a corresponding inverse inequality for B-splines of maximum smoothness have been shown, both with constants independent of the spline degree. We use these results to construct multigrid solvers for discretizations based on B-splines with maximum smoothness which exhibit robust convergence rates.

Clemens Hofreither, Stefan Takacs

We present a robust and efficient multigrid method for single-patch isogeometric discretizations using tensor product B-splines of maximum smoothness. Our method is based on a stable splitting of the spline space into a large subspace of interior splines which satisfy a robust inverse inequality, as well as one or several smaller subspaces which capture the boundary effects responsible for the spectral outliers which occur in Isogeometric Analysis. We then construct a multigrid smoother based on an additive subspace correction approach, applying a different smoother to each of the subspaces. For the interior splines, we use a mass smoother, whereas the remaining components are treated with suitably chosen Kronecker product smoothers or direct solvers. We prove that the resulting multigrid method exhibits iteration numbers which are robust with respect to the spline degree and the mesh size. Furthermore, it can be efficiently realized for discretizations of problems in arbitrarily high geometric dimension. Some numerical examples illustrate the theoretical results and show that the iteration numbers also scale relatively mildly with the problem dimension.