An Adaptive Algorithm Based on Stochastic Discontinuous Galerkin for Convection Dominated Equations with Random Data

TL;DR

This paper introduces an adaptive stochastic Galerkin method with mesh refinement and polynomial degree adaptation for solving convection dominated equations with random data, providing reliable error estimation and balanced spatial-stochastic error control.

Contribution

It develops a residual-based error estimator for an adaptive SIPG method applied to stochastic convection equations, including error bounds and error balancing strategies.

Findings

The error estimator is reliable and provides an upper bound for the energy error.

Numerical examples demonstrate effective error control and adaptivity in various random parameter scenarios.

The method successfully handles discontinuous and jump discontinuous diffusivity parameters.

Abstract

In this paper, we propose an adaptive approach, based on mesh refinement or parametric enrichment with polynomial degree adaption, for numerical solution of convection dominated equations with random input data. A parametric system emerged from an application of stochastic Galerkin approach is discretized by using symmetric interior penalty Galerkin (SIPG) method with upwinding for the convection term in the spatial domain. We derive a residual-based error estimator contributed by the error due to the SIPG discretization, the (generalized) polynomial chaos discretization in the stochastic space, and data oscillations. Then, the reliability of the proposed error estimator, an upper bound for the energy error up to a multiplicative constant, is shown. Moreover, to balance the errors stemmed from spatial and stochastic spaces, the truncation error emerged from Karhunen--Lo\`{e}ve expansion are considered in the numerical simulations. Last, several benchmark examples including a random diffusivity parameter, a random convectivity parameter, random diffusivity/convectivity parameters, and a random (jump) discontinuous diffusivity parameter, are tested to illustrate the performance of the proposed estimator.