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

Abstract

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 themore » 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.« less

Authors:

Pazner, Will  ^[1];

• Search DOE PAGES for author "Pazner, Will"
• Search DOE PAGES for ORCID "0000-0003-4885-2934"

Kolev, Tzanio  ^[2];

• Search DOE PAGES for author "Kolev, Tzanio"
• Search DOE PAGES for ORCID "0000-0002-2810-3090"

Dohrmann, Clark R. ^[3]

---

+ Show Author Affiliations

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)

Publication Date:

Wed Apr 26 00:00:00 EDT 2023

Research Org.:

Lawrence Livermore National Laboratory (LLNL), Livermore, CA (United States)

Sponsoring Org.:

USDOE National Nuclear Security Administration (NNSA); USDOE Laboratory Directed Research and Development (LDRD) Program

OSTI Identifier:

1987610

Report Number(s):

LLNL-JRNL-831792
Journal ID: ISSN 1064-8275; 1048960

Grant/Contract Number:  

AC52-07NA27344; NA0003525; 20-ERD-002

Resource Type:

Accepted Manuscript

Journal Name:

SIAM Journal on Scientific Computing

Additional Journal Information:

Journal Volume: 45; Journal Issue: 2; Journal ID: ISSN 1064-8275

Publisher:

Society for Industrial and Applied Mathematics (SIAM)

Country of Publication:

United States

Language:

English

Subject:

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