Abstract
The paper introduces a novel, hierarchical preconditioner based on nested dissection and hierarchical matrix compression. The preconditioner is intended for continuous and discontinuous Galerkin formulations of elliptic problems. We exploit the property that Schur complements arising in such problems can be well approximated by hierarchical matrices. An approximate factorization can be computed matrix-free and in a (quasi-)linear number of operations. The nested dissection is specifically designed to aid the factorization process using hierarchical matrices. We demonstrate the viability of the preconditioner on a range of 2D problems, including the Helmholtz equation and the elastic wave equation. Throughout all tests, including wave phenomena with high wavenumbers, the generalized minimal residual method (GMRES) with the proposed preconditioner converges in a very low number of iterations. We demonstrate that this is due to the hierarchical nature of our approach which makes the high wavenumber limit manageable.
Highlights
Many engineering problems require the solution of increasingly large linear systems of equations (1.1)Ax = b, where A \in \BbbR n\times n is an invertible matrix and x, b \in \BbbR n
We have presented a hierarchical preconditioning technique based on nested dissection and accelerated matrix arithmetic using hierarchical matrices
We can form the preconditioner matrix-free and in quasilinear \scrO (k2n log n) complexity, as well as apply it in \scrO complexity. This approach is truly general as it only requires the original matrix A and a nested dissection type elimination tree \scrE, which satisfies the well-separated property. The effectiveness of this approach has been demonstrated on a variety of elliptic problems, including wave problems in the high wavenumber limit
Summary
Many engineering problems require the solution of increasingly large linear systems of equations (1.1). A is sparse and arises from a finite element type discretization of a partial differential equation (PDE) on complex domains involving a prohibitively large number of degrees of freedom. This makes the application of direct solvers intractable, as fill-in often becomes excessive and memory requirements cannot be met [10]. Preconditioners are often required to keep the number of iterations to a minimum, as it may otherwise become a bottleneck This is especially true for time-dependent problems, where many solutions have to be computed. \ast Submitted to the journal's Methods and Algorithms for Scientific Computing section September 9, 2020; accepted for publication (in revised form) October 4, 2021; published electronically January 27, 2022
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