PRECONDITIONING OF INDEFINITE AND ALMOST SINGULAR FINITE-ELEMENT ELLIPTIC-EQUATIONS

This paper deals with two ways of solving discreti ed finite element e lliptic equations with indefinite and almost singular matrices. Such p roblems typically arise when applying the shifted inverse power iterat ion (SIPI) method to the generali ed eigenvalue problem Au = lambda Bu where A is defined from some discreti ed equation by a finite element self-adjoint coercive second-order elliptic operator and B is a mass- matrix operator. Both methods explore two-by-two block partitioning of the given matrices. One of the main matrix blocks corresponds to a co arse space (or, equivalently, to coarse-grid degrees of freedom) and c ontains the main singularity of the original problem. The other major block, possibly indefinite, is not as close to singular as the origina l matrix and is inverted by a preconditioned minimal residual (MINRES) method in the first method. The second method, which we use for compa rison, exploits preconditioned MINRES iterations for the reduced probl em obtained by eliminating the block unknowns that correspond to the c oarse discreti ation space. Here each iteration involves solving a coa rse-grid problem. Numerical examples are presented that demonstrate th e various aspects of both approaches for solving generali ed eigenvalu e problems applied to second-order finite element elliptic equations.