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Matrix‐free preconditioning for high‐order H (curl) discretizations
Author(s) -
Barker Andrew T.,
Kolev Tzanio
Publication year - 2021
Publication title -
numerical linear algebra with applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.02
H-Index - 53
eISSN - 1099-1506
pISSN - 1070-5325
DOI - 10.1002/nla.2348
Subject(s) - preconditioner , mathematics , finite element method , curl (programming language) , discretization , solver , condition number , matrix (chemical analysis) , tensor product , linear system , mathematical optimization , mathematical analysis , pure mathematics , computer science , eigenvalues and eigenvectors , physics , materials science , quantum mechanics , composite material , thermodynamics , programming language
Abstract The greater arithmetic intensity of high‐order finite element discretizations makes them attractive for implementation on next‐generation hardware, but assembly of high‐order finite element operators as matrices is prohibitively expensive. As a result, the development of general algebraic solvers for such operators has been an open research challenge. Fast matrix‐free application of high‐order operators has received significant attention in the literature in the context of Poisson‐type problems, but preconditioners and solvers for inverting more general operators are not very well‐developed. In this paper, we consider the problem of preconditioning a definite Maxwell operator at high polynomial order without assembling a matrix. We show that given efficient preconditioners for high‐order H 1 finite element problems on the same mesh, efficient H (curl) preconditioners can be constructed in an auxiliary space framework. We demonstrate the resulting preconditioners in a practical setting with tensor‐product basis functions on an unstructured mesh of quadrilaterals. Our approach uses a sparsified H 1 solver constructed on a low‐order mesh of the nodal points of the underlying high‐order space, and we show that the resulting H (curl) preconditioner is effective at very high polynomial orders for two‐dimensional model problems with complicated geometry, varying piecewise constant coefficients, and curved elements. The resulting preconditioner scales with nearly optimal O ( p d + 1 ) floating point operation count and optimal O ( p d ) memory transfer requirements, outperforming existing Maxwell preconditioners in the high‐order regime.