Solution Methods for Nonsymmetric Linear Systems with Large off-Diagonal Elements and Discontinuous Coefficients

Linear systems with very large off-diagonal elements and discontinuous coefficients (LODC systems) arise in some modeling cases, such as those involving heterogeneous media. Such problems are usually solved by domain decomposition methods, but these can be difficult to implement on unstructured grids or when the boundaries between subdomains have a complicated geometry. Gordon and Gordon have shown that Björck and Elfving’s (sequential) CGMN algorithm and their own block-parallel CARP-CG are very robust and efficient on strongly convection dominated cases (but without discontinuous coefficients). They have also shown that scaling the equations by dividing each equation by the L2-norm of its coefficients, called “geometric row scaling” (GRS), improves the convergence properties of Bi-CGSTAB and GMRES on nonsymmetric systems with discontinuous coefficients, provided the convection terms are only small to moderate. Given a system Ax = b, it is shown that if C is obtained from A by applying GRS, then the diagonal elements of CCT are larger than the off-diagonal ones, so the normal equations system is manageable. These two operations are inherent in the Kaczmarz algorithm, and hence also in CGMN and CARP-CG (which are CG-accelerations of Kaczmarz). It is shown that these two methods are also very effective on systems with discontinuous coefficients derived from strongly convection dominated elliptic PDEs. CGNR and CGNE also benefit greatly from this approach, but they are much less efficient.