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

A Numerical Approach to Space-Time Finite Elements for the Wave Equation

We study a space-time finite element approach for the nonhomogeneous wave equation using a continuous time Galerkin method. We present fully implicit examples in 1+1, 2+1, and 3+1 dimensions using linear quadrilateral, hexahedral, and tesseractic elements. Krylov solvers with additive Schwarz preconditioning are used for solving the linear system. We introduce a time decomposition strategy in preconditioning which significantly improves performance when compared with unpreconditioned cases.Comment: 9 pages, 5 figures, 5 table

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

Space-Time Domain Decomposition Methods for Diffusion Problems in Mixed Formulations

This paper is concerned with global-in-time, nonoverlapping domain decomposition methods for the mixed formulation of the diffusion problem. Two approaches are considered: one uses the time-dependent Steklov-Poincar\'e operator and the other uses Optimized Schwarz Waveform Relaxation (OSWR) based on Robin transmission conditions. For each method, a mixed formulation of an interface problem on the space-time interfaces between subdomains is derived, and different time grids are employed to adapt to different time scales in the subdomains. Demonstrations of the well-posedness of the subdomain problems involved in each method and a convergence proof of the OSWR algorithm are given for the mixed formulation. Numerical results for 2D problems with strong heterogeneities are presented to illustrate the performance of the two methods