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Title: Low-Order Preconditioning for the High-Order Finite Element de Rham Complex

Journal Article · · SIAM Journal on Scientific Computing
DOI: https://doi.org/10.1137/22m1486534 · OSTI ID:1987610
ORCiD logo [1]; ORCiD logo [2];  [3]
  1. Portland State Univ., OR (United States); Lawrence Livermore National Laboratory (LLNL), Livermore, CA (United States)
  2. Lawrence Livermore National Laboratory (LLNL), Livermore, CA (United States)
  3. Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
+ Show Author Affiliations

Here, we present a unified framework for constructing spectrally equivalent low-order-refined discretizations for the high-order finite element de Rham complex. This theory covers diffusion problems in H1, H(curl), and H(div) and is based on combining a low-order discretization posed on a refined mesh with a high-order basis for Nédélec and Raviart–Thomas elements that makes use of the concept of polynomial histopolation (polynomial fitting using prescribed mean values over certain regions). This spectral equivalence, coupled with algebraic multigrid methods constructed using the low-order discretization, results in highly scalable matrix-free preconditioners for high-order finite element problems in the full de Rham complex. Additionally, a new lowest-order (piecewise constant) preconditioner is developed for high-order interior penalty discontinuous Galerkin (DG) discretizations, for which spectral equivalence results and convergence proofs for algebraic multigrid methods are provided. In all cases, the spectral equivalence results are independent of polynomial degree and mesh size; for DG methods, they are also independent of the penalty parameter. These new solvers are flexible and easy to use; any “black-box” preconditioner for low-order problems can be used to create an effective and efficient preconditioner for the corresponding high-order problem. A number of numerical experiments are presented, based on an implementation in the finite element library MFEM. A range of challenging three-dimensional problems are used to corroborate the theoretical properties and demonstrate the flexibility and scalability of the method.

Research Organization:
Lawrence Livermore National Laboratory (LLNL), Livermore, CA (United States)
Sponsoring Organization:
USDOE National Nuclear Security Administration (NNSA); USDOE Laboratory Directed Research and Development (LDRD) Program
Grant/Contract Number:
AC52-07NA27344; NA0003525; 20-ERD-002
OSTI ID:
1987610
Report Number(s):
LLNL-JRNL-831792; 1048960
Journal Information:
SIAM Journal on Scientific Computing, Vol. 45, Issue 2; ISSN 1064-8275
Publisher:
Society for Industrial and Applied Mathematics (SIAM)Copyright Statement
Country of Publication:
United States
Language:
English

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Figures / Tables (15)

Fig. 2.1(p. 5)figure Fig. 2.1
Fig. 3.1(p. 7)figure Fig. 3.1
Fig. 4.1(p. 10)figure Fig. 4.1
Fig. 6.1(p. 17)figure Fig. 6.1
Fig. 6.2(p. 18)figure Fig. 6.2
Table 6.1(p. 18)table Table 6.1
Fig. 6.3(p. 19)figure Fig. 6.3
Table 6.2(p. 19)table Table 6.2
Fig. 6.4(p. 20)figure Fig. 6.4
Table 6.3(p. 20)table Table 6.3
Table 6.4(p. 21)table Table 6.4
Table 6.5(p. 21)table Table 6.5
Fig. 6.5(p. 22)figure Fig. 6.5
Table 6.6(p. 22)table Table 6.6
Fig. 6.6(p. 23)figure Fig. 6.6

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97 MATHEMATICS AND COMPUTING
preconditioning
high-order
finite elements
de Rham complex
algebraic multigrid