Corpus ID: 238583186

A scalable and robust vertex-star relaxation for high-order FEM

  title={A scalable and robust vertex-star relaxation for high-order FEM},
  author={Pablo D Brubeck and Patrick E. Farrell},
Pavarino proved that the additive Schwarz method with vertex patches and a loworder coarse space gives a p-robust solver for symmetric and coercive problems [33]. However, for very high polynomial degree it is not feasible to assemble or factorize the matrices for each patch. In this work we introduce a direct solver for separable patch problems that scales to very high polynomial degree on tensor product cells. The solver constructs a tensor product basis that diagonalizes the blocks in the… Expand


Fast Tensor Product Schwarz Smoothers for High-Order Discontinuous Galerkin Methods
The inversion of matrices associated to mesh cells and to the patches around a vertex is studied in order to obtain fast local solvers for additive and multiplicative subspace correction methods for high-order discontinuous Galerkin (DG) finite element methods. Expand
PCPATCH: software for the topological construction of multigrid relaxation methods
This paper presents a unifying software abstraction, PCPATCH, for the topological construction of space decompositions for multigrid relaxation methods, and facilitates the elegant expression of a wide range of schemes merely by varying solver options at runtime. Expand
Efficient low-order refined preconditioners for high-order matrix-free continuous and discontinuous Galerkin methods
  • Will Pazner
  • Computer Science, Mathematics
  • SIAM J. Sci. Comput.
  • 2020
This paper designs preconditioners for the matrix-free solution of high-order continuous and discontinuous Galerkin discretizations of elliptic problems based on FEM-SEM equivalence and additive Schwarz methods and presents results on a variety of examples. Expand
Scalable Low-Order Finite Element Preconditioners for High-Order Spectral Element Poisson Solvers
Low-order finite element (FE) systems are considered as preconditioners for spectral element (SE) discretizations of the Poisson problem in canonical and complex domains. The FE matrices are based ...
Scalable tensor-product preconditioners for high-order finite-element methods: Scalar equations
A tensor-product-based preconditioner for high-order discontinuous-Galerkin (DG) discretizations, based on approximating the block diagonal of the Jacobian matrix corresponding to element-wise coupling with the sum of tensor products of small one-dimensional matrices, which is effective for solving 4D (3D-space+time) DG discretization problems up to 32nd order. Expand
An Additive Variant of the Schwarz Alternating Method for the Case of Many Subregions
In recent years, there has been a revival of interest in the Schwarz alternating method. Other domain decomposition algorithms, in particular the so called iterative substructuring methods, have alsoExpand
Approximate tensor-product preconditioners for very high order discontinuous Galerkin methods
A new tensor-product based preconditioner for discontinuous Galerkin methods with polynomial degrees higher than those typically employed, combined with a matrix-free Newton–Krylov solver allows for the solution of DG systems in linear time in p per degree of freedom in 2D, and reduces the computational complexity in 3D. Expand
Interior penalty tensor-product preconditioners for high-order discontinuous Galerkin discretizations
In this paper, we introduce an interior penalty tensor-product preconditioner for the implicit time integration of discontinuous Galerkin discretizations of partial differential equations withExpand
GPU accelerated spectral finite elements on all-hex meshes
A spectral element finite element scheme that efficiently solves elliptic problems on unstructured hexahedral meshes using a matrix-free preconditionsed conjugate gradient algorithm and an additive Schwartz two-scale preconditioner that allows h-independence convergence. Expand
Polynomial robust stability analysis for $H(\textrm{div})$-conforming finite elements for the Stokes equations
In this work we consider a discontinuous Galerkin method for the discretization of the Stokes problem. We use $H(\textrm{div})$-conforming finite elements as they provide major benefits such as exactExpand