Sparse Hierarchical Preconditioners Using Piecewise Smooth Approximations of Eigenvectors
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When solving linear systems arising from PDE discretizations, iterative methods (such as Conjugate Gradient, GMRES, or MINRES) are often the only practical choice. To converge in a small number of iterations, however, they have to be coupled with an efficient preconditioner. The efficiency of the preconditioner depends largely on its accuracy on the eigenvectors corresponding to small eigenvalues, and unfortunately, black-box methods typically cannot guarantee sufficient accuracy on these eigenvectors. Thus, constructing the preconditioner becomes a very problem-dependent task. We describe a hierarchical approximate factorization approach which addresses this issue by focusing on improving the accuracy on smooth eigenvectors (such eigenvectors typically correspond to the small eigenvalues). The improved accuracy is achieved by preserving the action of the factorized matrix on piecewise polynomial functions of the PDE domain. Based on the factorization, we propose a family of sparse preconditioners with O(n) or O( n n) construction complexities. Our methods exhibit the optimal O(n ) solution times in benchmarks run on large elliptic problems of different types, arising for example in flow or mechanical simulations. In the case of the linear elasticity equation the preconditioners are exact on the near-kernel rigid body modes.
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