Space Decompositions and Solvers for Discontinuous Galerkin Methods

We present a brief overview of the different domain and space decomposition techniques that enter in developing and analyzing solvers for discontinuous Galerkin methods. Emphasis is given to the novel and distinct features that arise when considering DG discretizations over conforming methods. Connections and differences with the conforming approaches are emphasized.Comment: 2 pages 2 figures no table

Nonoverlapping domain decomposition preconditioners for discontinuous Galerkin finite element methods in H2H^2-type norms

We analyse the spectral bounds of nonoverlapping domain decomposition preconditioners for $hp$-version discontinuous Galerkin finite element methods in $H^2$-type norms, which arise in applications to fully nonlinear Hamilton--Jacobi--Bellman partial differential equations. We show that for a symmetric model problem, the condition number of the preconditioned system is at most of order $1+ p^6 H^3 /q^3 h^3$, where $H$ and $h$ are respectively the coarse and fine mesh sizes, and $q$ and $p$ are respectively the coarse and fine mesh polynomial degrees. This represents the first result for this class of methods that explicitly accounts for the dependence of the condition number on $q$, and its sharpness is shown numerically. The key analytical tool is an original optimal order approximation result between fine and coarse discontinuous finite element spaces.\ud \ud We then go beyond the model problem and show computationally that these methods lead to efficient and competitive solvers in practical applications to nonsymmetric, fully nonlinear Hamilton--Jacobi--Bellman equations

Nonoverlapping domain decomposition preconditioners for discontinuous Galerkin approximations of Hamilton--Jacobi--Bellman equations

We analyse a class of nonoverlapping domain decomposition preconditioners for nonsymmetric linear systems arising from discontinuous Galerkin finite element approximation of fully nonlinear Hamilton--Jacobi--Bellman (HJB) partial differential equations. These nonsymmetric linear systems are uniformly bounded and coercive with respect to a related symmetric bilinear form, that is associated to a matrix $\mathbf{A}$. In this work, we construct a nonoverlapping domain decomposition preconditioner $\mathbf{P}$, that is based on $\mathbf{A}$, and we then show that the effectiveness of the preconditioner for solving the} nonsymmetric problems can be studied in terms of the condition number $\kappa(\mathbf{P}^{-1}\mathbf{A})$. In particular, we establish the bound $\kappa(\mathbf{P}^{-1}\mathbf{A}) \lesssim 1+ p^6 H^3 /q^3 h^3$, where $H$ and $h$ are respectively the coarse and fine mesh sizes, and $q$ and $p$ are respectively the coarse and fine mesh polynomial degrees. This represents the first such result for this class of methods that explicitly accounts for the dependence of the condition number on $q$; our analysis is founded upon an original optimal order approximation result between fine and coarse discontinuous finite element spaces. Numerical experiments demonstrate the sharpness of this bound. Although the preconditioners are not robust with respect to the polynomial degree, our bounds quantify the effect of the coarse and fine space polynomial degrees. Furthermore, we show computationally that these methods are effective in practical applications to nonsymmetric, fully nonlinear HJB equations under $h$-refinement for moderate polynomial degrees

A Combined Preconditioning Strategy for Nonsymmetric Systems

We present and analyze a class of nonsymmetric preconditioners within a normal (weighted least-squares) matrix form for use in GMRES to solve nonsymmetric matrix problems that typically arise in finite element discretizations. An example of the additive Schwarz method applied to nonsymmetric but definite matrices is presented for which the abstract assumptions are verified. A variable preconditioner, combining the original nonsymmetric one and a weighted least-squares version of it, is shown to be convergent and provides a viable strategy for using nonsymmetric preconditioners in practice. Numerical results are included to assess the theory and the performance of the proposed preconditioners.Comment: 26 pages, 3 figure

Convergence of HX Preconditioner for Maxwell's Equations with Jump Coefficients (ii): The Main Results

This paper is the second one of two serial articles, whose goal is to prove convergence of HX Preconditioner (proposed by Hiptmair and Xu, 2007) for Maxwell's equations with jump coefficients. In this paper, based on the auxiliary results developed in the first paper (Hu, 2017), we establish a new regular Helmholtz decomposition for edge finite element functions in three dimensions, which is nearly stable with respect to a weight function. By using this Helmholtz decomposition, we give an analysis of the convergence of the HX preconditioner for the case with strongly discontinuous coefficients. We show that the HX preconditioner possesses fast convergence, which not only is nearly optimal with respect to the finite element mesh size but also is independent of the jumps in the coefficients across the interface between two neighboring subdomains.Comment: with 25 pages, 2 figure