Additive Schwarz for Anisotropic Elliptic Problems

Abstract

Strong anisotropic behavior is known to cause difficulties for many standard iterative methods when solving the algebraic equations resulting from a discrete approximation of a partial differential equation. We present a robust domain decomposition algorithm that will work reliably also under such conditions. The method is derived and discussed on a model region, but our construction shows what may be needed in the general case. Our coarse space corresponds somewhat to the more complex relaxation that is often used in multigrid algorithms under similar conditions. The condition number of our iteration operator is proportional to H/h, the ratio of the subdomain diameter to the element diameter in the discrete problem. This quantity is independent of the coefficients, that is, their discontinuities and their anisotropy. The implementation confirms the good parallel properties characteristic for this class of iterative methods. The paper concludes with numerical examples confirming the theoretical result.

Supported in part by The Institute for Mathematics and its Applications, University of Minnesota and in part by The Norwegian Research Council, grant 113492/420.

Supported in part by National Science Foundation Grant NSF-CCR-9503408 and Polish Scientific Grant 2P03A00909.