A Preconditioned Conjugate Gradient Method for Eigenvalue Problems and its Implementation in a Subspace

Abstract — We treat systematically the preconditioned steepest ascent method and the preconditioned conjugate gradient method for eigenvalue problems and present convergence rate estimates. We also suggest a modification of the methods, that makes it possible to implement them in a subspace (such as that of mesh functions, defined in the mesh-points on the dividing line for the domain decomposition methods). We discuss as an example an eigenvalue problem for -Δ h (a mesh discretisation of Laplacian) and show that the rate of convergence does not slow as h → 0.

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