Fast Tensor Product Schwarz Smoothers for High-Order Discontinuous Galerkin Methods

We discuss the efficient implementation of powerful domain decomposition smoothers for multigrid methods for high-order discontinuous Galerkin (DG) finite element methods. In particular, we study the inversion of matrices associated to mesh cells and to the patches around a vertex, respectively, in order to obtain fast local solvers for additive and multiplicative subspace correction methods. The effort of inverting local matrices for tensor product polynomials of degree k is reduced from {\mathcal{O}(k^{3d})} to {\mathcal{O}(dk^{d+1})} by exploiting the separability of the differential operator and resulting low rank representation of its inverse as a prototype for more general low rank representations in space dimension d.

A Semi-Uniform Multigrid Algorithm for Solving Elliptic Interface Problems

We introduce a new geometric multigrid algorithm to solve elliptic interface problems. First we discretize the problems by the usual {P_{1}}-conforming finite element methods on a semi-uniform grid which is obtained by refining a uniform grid. To solve the algebraic system, we adopt subspace correction methods for which we use uniform grids as the auxiliary spaces. To enhance the efficiency of the algorithms, we define a new transfer operator between a uniform grid and a semi-uniform grid so that the transferred functions satisfy the flux continuity along the interface. In the auxiliary space, the system is solved by the usual multigrid algorithm with a similarly modified prolongation operator. We show {\mathcal{W}}-cycle convergence for the proposed multigrid algorithm. We demonstrate the performance of our multigrid algorithm for problems having various ratios of parameters. We observe that the computational complexity of our algorithms are robust for all problems we tested.

ROBUST SUBSPACE CORRECTION METHODS FOR NEARLY SINGULAR SYSTEMS

In this paper, we discuss convergence results for general (successive) subspace correction methods for solving nearly singular systems of equations. We provide parameter independent estimates under appropriate assumptions on the subspace solvers and space decompositions. The main assumption is that any component in the kernel of the singular part of the system can be decomposed into a sum of local (in each subspace) kernel components. This assumption also covers the case of "hidden" nearly singular behavior due to decreasing mesh size in the systems resulting from finite element discretizations of second order elliptic problems. To illustrate our abstract convergence framework, we analyze a multilevel method for the Neumann problem (H(grad) system), and also two-level methods for H(div) and H(curl) systems.