A matrixfree ILU realization based on surrogates
Abstract
Matrixfree techniques play an increasingly important role in largescale simulations. Schur complement techniques and massively parallel multigrid solvers for secondorder elliptic partial differential equations can significantly benefit from reduced memory traffic and consumption. The matrixfree approach often restricts solver components to purely local operations, for instance, the Jacobi or GaussSeidelSmoothers in multigrid methods. An incomplete LU (ILU) decomposition cannot be calculated from local information and is therefore not amenable to an onthefly computation which is typically needed for matrixfree calculations. It generally requires the storage and factorization of a sparse matrix which contradicts the low memory requirements in large scale scenarios. In this work, we propose a matrixfree ILU realization. More precisely, we introduce a memoryefficient, matrixfree ILU(0)Smoother component for loworder conforming finite elements on tetrahedral hybrid grids. Hybrid grids consist of an unstructured macromesh which is subdivided into a structured micromesh. The ILU(0) is used for degreesoffreedom assigned to the interior of macrotetrahedra. This ILU(0)Smoother can be used for the efficient matrixfree evaluation of the SteklovPoincare operator from domaindecomposition methods. After introducing and formally defining our smoother, we investigate its performance on refined macrotetrahedra. Secondly, the ILU(0)Smoother on the macrotetrahedrons is implemented via surrogate matrix polynomials in conjunction with a fast onthefly evaluation scheme resulting in an efficient matrixfree algorithm. The polynomial coefficients are obtained by solving a leastsquares problem on a small part of the factorized ILU(0) matrices to stay memory efficient. The convergence rates of this smoother with respect to the polynomial order are thoroughly studied.
 Publication:

arXiv eprints
 Pub Date:
 October 2022
 DOI:
 10.48550/arXiv.2210.15280
 arXiv:
 arXiv:2210.15280
 Bibcode:
 2022arXiv221015280D
 Keywords:

 Mathematics  Numerical Analysis