Abstract
We consider the approximation of the inverse of the finite element stiffness matrix in the data sparse {mathcal{H}}-matrix format. For a large class of shape regular but possibly non-uniform meshes including algebraically graded meshes, we prove that the inverse of the stiffness matrix can be approximated in the {mathcal{H}}-matrix format at an exponential rate in the block rank. Since the storage complexity of the hierarchical matrix is logarithmic-linear and only grows linearly in the block-rank, we obtain an efficient approximation that can be used, e.g., as an approximate direct solver or preconditioner for iterative solvers.
Highlights
Discretizations of elliptic partial differential equations on a domain Ω ⊆ Rd using the classical finite element method (FEM) usually produce sparse linear systems of equations Ax = b with storage requirements linear in the number of unknowns and linear complexity for the matrix-vector multiplication
A drawback of these methods is that convergence can be slow for matrices with large condition numbers unless a suitable preconditioner is employed. These preconditioners have to be taylored to the problem at hand making black box preconditioners that are based on direct solvers interesting
Our main result states that the inverses of FEM matrices for such meshes can be approximated by hierarchical matrices such that the error converges exponentially in the H-matrix block rank r
Summary
A drawback of these methods is that convergence can be slow for matrices with large condition numbers unless a suitable preconditioner is employed These preconditioners have to be taylored to the problem at hand making black box preconditioners that are based on (approximate) direct solvers interesting. Using an H-matrix approximation to the inverse A−1 gives an approximate direct solution method of logarithmic-linear complexity that can be applied efficiently to multiple right-hand sides. In order to explain the numerical success of these approximations, first observed in [19], several works in the literature provide existence results of approximations to the inverse matrices in the H-matrix format. Our main result states that the inverses of FEM matrices for such meshes can be approximated by hierarchical matrices such that the error converges exponentially in the H-matrix block rank r. ) on ⊆T aandm⋃eshsuTppTby(v)su⊆ppRTd( ,vw)∶h=ic{hTs∈ligTht|lyv|dTif≢fer0s} f.roInmptahretiucsuulaarl,dwefeinihtaiovne of a support, namely, supp(v)∶={x ∈ Ω | v(x) ≠ 0} ⊆ Rd
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