MINRES-QLP: A Krylov Subspace Method for Indefinite or Singular Symmetric Systems
SIAM Journal on Scientific Computing
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This paper deals with two ways of solving discretized finite element elliptic equations with indefinite and almost singular matrices. Such problems typically arise when applying the shifted inverse power iteration (SIPI) method to the generalized eigenvalue problem $A{\bf u}=\lambda\,B{\bf u}$ where A is defined from some discretized 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 coarse space (or, equivalently, to coarse-grid degrees of freedom) and contains the main singularity of the original problem. The other major block, possibly indefinite, is not as close to singular as the original matrix and is inverted by a preconditioned minimal residual (MINRES) method in the first method. The second method, which we use for comparison, exploits preconditioned MINRES iterations for the reduced problem obtained by eliminating the block unknowns that correspond to the coarse discretization space. Here each iteration involves solving a coarse-grid problem. Numerical examples are presented that demonstrate the various aspects of both approaches for solving generalized eigenvalue problems applied to second-order finite element elliptic equations.