Low-Order Preconditioning for the High-Order Finite Element de Rham Complex
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
Similar Records
Efficient Low-Order Refined Preconditioners for High-Order Matrix-Free Continuous and Discontinuous Galerkin Methods
Diffusion Synthetic Acceleration for High Order S{sub N} Transport on Meshes with Curved Surfaces