Subspace Correction Methods for discontinuous Galerkin approximations
Prof. Ludmil ZikatanovNov. 4, 2008, 3:30 p.m. MZ 005A
In this work we employ the subspace correction framework to design eﬃcient iterative solution methods for the linear systems resulting from discontinuous Galerkin (DG) discretizations. Subspace correction methods refer to a large class of algorithms used in scientiﬁc and engineering computing, based on a divide and conquer strategies. Many iterative methods (simple or complicated, traditional or modern) fall into this category. Examples include the classical Jacobi, Gauss-Seidel and SOR methods, and also multigrid and domain decomposition methods.
The focus of our presentation will be on the design of new iterative techniques for DG based on special subspace splittings. The key construction is a simple orthogonal (in energy inner product) decomposition of the discontinuous ﬁnite element space, into sum of the well-known Crouziex-Raviart (CR) ﬁnite element space and its complementary space. Our algorithms can be regarded as special two-grid methods and include two basic steps: (1) local relaxation for the solution of a well-conditioned problem (damping the error components in the complementary spaces mentioned above); (2) solution of a linear system corresponding to more conforming (CR) discretization (resolving global error components).
The talk is based on joint work with B. Ayuso from Universidad Autonoma de Madrid, (Spain) and on an an ongoing joint work with J. Kraus and I. Georgiev from RICAM (Austria).