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

Lower Bounds in Domain Decomposition

An important indicator of the efficiency of a domain decomposition preconditioner is the condition number of the preconditioned system. Upper bounds for the condition numbers of the preconditioned systems have been the focus of most analyses in domain decomposition [21, 20, 23]. However, in order to have a fair comparison of two preconditioners, the sharpness of the respective upper bounds must first be established, which means that we need to derive lower bounds for the condition numbers of the preconditioned systems

A FETI-DP TYPE DOMAIN DECOMPOSITION ALGORITHM FOR THREE-DIMENSIONAL INCOMPRESSIBLE STOKES EQUATIONS

The FETI-DP (dual-primal finite element tearing and interconnecting) algorithms, proposed by the authors in [SIAM J. Numer. Anal., 51 (2013), pp. 1235–1253] and [Internat. J. Numer. Methods Engrg., 94 (2013), pp. 128–149] for solving incompressible Stokes equations, are extended to three-dimensional problems. A new analysis of the condition number bound for using the Dirichlet preconditioner is given. The algorithm and analysis are valid for mixed finite elements with both continuous and discontinuous pressures. An advantage of this new analysis is that the numerous coarse level velocity components, required in the previous analysis to enforce the divergence-free subdomain boundary velocity conditions, are no longer needed. This greatly reduces the size of the coarse level problem in the algorithm, especially for three-dimensional problems. The coarse level velocity space can be chosen as simple as those coarse spaces for solving scalar elliptic problems corresponding to each velocity component. Both the Dirichlet and lumped preconditioners are analyzed using the same framework in this new analysis. Their condition number bounds are proved to be independent of the number of subdomains for fixed subdomain problem size. Numerical experiments in both two and three dimensions, using mixed finite elements with both continuous and discontinuous pressures, demonstrate the convergence rate of the algorithms

A class of alternate strip-based domain decomposition methods for elliptic partial differential Equations

The domain decomposition strategies proposed in this thesis are efficient preconditioning techniques with good parallelism properties for the discrete systems which arise from the finite element approximation of symmetric elliptic boundary value problems in two and three-dimensional Euclidean spaces. For two-dimensional problems, two new domain decomposition preconditioners are introduced, such that the condition number of the preconditioned system is bounded independently of the size of the subdomains and the finite element mesh size. First, the alternate strip-based (ASB2) preconditioner is based on the partitioning of the domain into a finite number of nonoverlapping strips without interior vertices. This preconditioner is obtained from direct solvers inside the strips and a direct fast Poisson solver on the edges between strips, and contains two stages. At each stage the strips change such that the edges between strips at one stage are perpendicular on the edges between strips at the other stage. Next, the alternate strip-based substructuring (ASBS2) preconditioner is a Schur complement solver for the case of a decomposition with multiple nonoverlapping subdomains and interior vertices. The subdomains are assembled into nonoverlapping strips such that the vertices of the strips are on the boundary of the given domain, the edges between strips align with the edges of the subdomains and their union contains all of the interior vertices of the initial decomposition. This preconditioner is produced from direct fast Poisson solvers on the edges between strips and the edges between subdo- mains inside strips, and also contains two stages such that the edges between strips at one stage are perpendicular on the edges between strips at the other stage. The extension to three-dimensional problems is via solvers on slices of the domain

A fully algebraic and robust two-level Schwarz method based on optimal local approximation spaces

Two-level domain decomposition preconditioners lead to fast convergence and scalability of iterative solvers. However, for highly heterogeneous problems, where the coefficient function is varying rapidly on several possibly non-separated scales, the condition number of the preconditioned system generally depends on the contrast of the coefficient function leading to a deterioration of convergence. Enhancing the methods by coarse spaces constructed from suitable local eigenvalue problems, also denoted as adaptive or spectral coarse spaces, restores robust, contrast-independent convergence. However, these eigenvalue problems typically rely on non-algebraic information, such that the adaptive coarse spaces cannot be constructed from the fully assembled system matrix. In this paper, a novel algebraic adaptive coarse space, which relies on the a-orthogonal decomposition of (local) finite element (FE) spaces into functions that solve the partial differential equation (PDE) with some trace and FE functions that are zero on the boundary, is proposed. In particular, the basis is constructed from eigenmodes of two types of local eigenvalue problems associated with the edges of the domain decomposition. To approximate functions that solve the PDE locally, we employ a transfer eigenvalue problem, which has originally been proposed for the construction of optimal local approximation spaces for multiscale methods. In addition, we make use of a Dirichlet eigenvalue problem that is a slight modification of the Neumann eigenvalue problem used in the adaptive generalized Dryja-Smith-Widlund (AGDSW) coarse space. Both eigenvalue problems rely solely on local Dirichlet matrices, which can be extracted from the fully assembled system matrix. By combining arguments from multiscale and domain decomposition methods we derive a contrast-independent upper bound for the condition number

Multispace and Multilevel BDDC

BDDC method is the most advanced method from the Balancing family of iterative substructuring methods for the solution of large systems of linear algebraic equations arising from discretization of elliptic boundary value problems. In the case of many substructures, solving the coarse problem exactly becomes a bottleneck. Since the coarse problem in BDDC has the same structure as the original problem, it is straightforward to apply the BDDC method recursively to solve the coarse problem only approximately. In this paper, we formulate a new family of abstract Multispace BDDC methods and give condition number bounds from the abstract additive Schwarz preconditioning theory. The Multilevel BDDC is then treated as a special case of the Multispace BDDC and abstract multilevel condition number bounds are given. The abstract bounds yield polylogarithmic condition number bounds for an arbitrary fixed number of levels and scalar elliptic problems discretized by finite elements in two and three spatial dimensions. Numerical experiments confirm the theory.Comment: 26 pages, 3 figures, 2 tables, 20 references. Formal changes onl

An agglomeration-based massively parallel non-overlapping additive Schwarz preconditioner for high-order discontinuous Galerkin methods on polytopic grids

In this article we design and analyze a class of two-level non-overlapping additive Schwarz preconditioners for the solution of the linear system of equations stemming from discontinuous Galerkin discretizations of second-order elliptic partial differential equations on polytopic meshes. The preconditioner is based on a coarse space and a non-overlapping partition of the computational domain where local solvers are applied in parallel. In particular, the coarse space can potentially be chosen to be non-embedded with respect to the finer space; indeed it can be obtained from the fine grid by employing agglomeration and edge coarsening techniques. We investigate the dependence of the condition number of the preconditioned system with respect to the diffusion coefficient and the discretization parameters, i.e., the mesh size and the polynomial degree of the fine and coarse spaces. Numerical examples are presented which confirm the theoretical bounds

Graph partitioning using matrix values for preconditioning symmetric positive definite systems

Prior to the parallel solution of a large linear system, it is required to perform a partitioning of its equations/unknowns. Standard partitioning algorithms are designed using the considerations of the efficiency of the parallel matrix-vector multiplication, and typically disregard the information on the coefficients of the matrix. This information, however, may have a significant impact on the quality of the preconditioning procedure used within the chosen iterative scheme. In the present paper, we suggest a spectral partitioning algorithm, which takes into account the information on the matrix coefficients and constructs partitions with respect to the objective of enhancing the quality of the nonoverlapping additive Schwarz (block Jacobi) preconditioning for symmetric positive definite linear systems. For a set of test problems with large variations in magnitudes of matrix coefficients, our numerical experiments demonstrate a noticeable improvement in the convergence of the resulting solution scheme when using the new partitioning approach