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

TL;DR

This paper develops a low-rank stochastic Galerkin discontinuous Galerkin method for convection diffusion equations with random coefficients, providing error estimates and demonstrating computational efficiency through numerical experiments.

Contribution

It introduces a low-rank Krylov subspace approach to efficiently solve high-dimensional stochastic Galerkin systems for convection diffusion equations.

Findings

Reduced computational complexity and storage due to low-rank methods.

Effective handling of stochastic convection diffusion problems with error control.

Numerical experiments confirm the efficiency and accuracy of the proposed approach.

Abstract

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.