Kronecker Product Approximation Preconditioners for Convection-diffusion Model Problems

We consider the iterative solution of the linear systems arising from four convection-diffusion model problems: the scalar convection-diffusion problem, Stokes problem, Oseen problem, and Navier-Stokes problem. We give the explicit Kronecker product structure of the coefficient matrices, especially the Kronecker product structure for the convection term. For the latter three model cases, the coefficient matrices have a $2 \times 2$ blocks, and each block is a Kronecker product or a summation of several Kronecker products. We use the Kronecker products and block structures to design the diagonal block preconditioner, the tridiagonal block preconditioner and the constraint preconditioner. We can find that the constraint preconditioner can be regarded as the modification of the tridiagonal block preconditioner and the diagonal block preconditioner based on the cell Reynolds number. That's the reason why the constraint preconditioner is usually better. We also give numerical examples to show the efficiency of this kind of Kronecker product approximation preconditioners

Matrix-equation-based strategies for convection-diffusion equations

We are interested in the numerical solution of nonsymmetric linear systems arising from the discretization of convection-diffusion partial differential equations with separable coefficients and dominant convection. Preconditioners based on the matrix equation formulation of the problem are proposed, which naturally approximate the original discretized problem. For certain types of convection coefficients, we show that the explicit solution of the matrix equation can effectively replace the linear system solution. Numerical experiments with data stemming from two and three dimensional problems are reported, illustrating the potential of the proposed methodology

Multilevel preconditioning based on discrete symmetrization for convection-diffusion equations

AbstractThe subject of this paper is an additive multilevel preconditioning approach for convection-diffusion problems. Our particular interest is in the convergence behavior for convection-dominated problems which are discretized by the streamline diffusion method. The multilevel preconditioner is based on a transformation of the discrete problem which reduces the relative size of the skew-symmetric part of the operator. For the constant coefficient case, an analysis of the convergence properties of this multilevel preconditioner is given in terms of its dependence on the convection size. Moreover, the results of computational experiments for more general convection-diffusion problems are presented and our new preconditioner is compared to standard multilevel preconditioning

Approximate tensor-product preconditioners for very high order discontinuous Galerkin methods

In this paper, we develop a new tensor-product based preconditioner for discontinuous Galerkin methods with polynomial degrees higher than those typically employed. This preconditioner uses an automatic, purely algebraic method to approximate the exact block Jacobi preconditioner by Kronecker products of several small, one-dimensional matrices. Traditional matrix-based preconditioners require $\mathcal{O}(p^{2d})$ storage and $\mathcal{O}(p^{3d})$ computational work, where $p$ is the degree of basis polynomials used, and $d$ is the spatial dimension. Our SVD-based tensor-product preconditioner requires $\mathcal{O}(p^{d+1})$ storage, $\mathcal{O}(p^{d+1})$ work in two spatial dimensions, and $\mathcal{O}(p^{d+2})$ work in three spatial dimensions. Combined with a matrix-free Newton-Krylov solver, these preconditioners allow for the solution of DG systems in linear time in $p$ per degree of freedom in 2D, and reduce the computational complexity from $\mathcal{O}(p^9)$ to $\mathcal{O}(p^5)$ in 3D. Numerical results are shown in 2D and 3D for the advection and Euler equations, using polynomials of degree up to $p=15$. For many test cases, the preconditioner results in similar iteration counts when compared with the exact block Jacobi preconditioner, and performance is significantly improved for high polynomial degrees $p$.Comment: 40 pages, 15 figure

Stochastic Discontinuous Galerkin Methods with Low--Rank Solvers for Convection Diffusion Equations

We investigate numerical behaviour of a convection diffusion equation with random coefficients by approximating statistical moments of the solution. Stochastic Galerkin approach, turning the original stochastic problem to a system of deterministic convection diffusion equations, is used to handle the stochastic domain in this study, whereas discontinuous Galerkin method is used to discretize spatial domain due to its local mass conservativity. A priori error estimates of the stationary problem and stability estimate of the unsteady model problem are derived in the energy norm. To address the curse of dimensionality of Stochastic Galerkin method, we take advantage of the low--rank Krylov subspace methods, which reduce both the storage requirements and the computational complexity by exploiting a Kronecker--product structure of system matrices. The efficiency of the proposed methodology is illustrated by numerical experiments on the benchmark problems.Comment: 50 pages, 9 figures, 9 table