Optimal error estimate of an isoparametric upwind discontinuous Galerkin method for radiation transport equation on curved domains

TL;DR

This paper establishes optimal error estimates for an isoparametric upwind discontinuous Galerkin method applied to the radiation transport equation on curved domains, addressing geometric and stability challenges.

Contribution

It introduces a novel analysis framework for the method's stability and error estimates on curved domains using isoparametric mapping and auxiliary operators.

Findings

Proves the bilinear form's continuity with respect to the DG norm.

Precisely estimates the geometric approximation error of inflow boundary.

Achieves optimal convergence rate in the DG norm.

Abstract

In recent years, high-order finite element methods on high-order meshes have attracted considerable attention. This work investigates the isoparametric upwind discontinuous Galerkin method for the radiation transport equation on a bounded domain with a piecewise $C^{k+1}$ smooth curved boundary. We use the isoparametric mapping to approximate the curved domain and construct a curved upwind discontinuous Galerkin scheme. The first-order hyperbolic nature and the complexity introduced by non-affine transformation, lead to additional difficulties for geometric approximation, numerical stability and the optimal error estimate. To address these issues, with the help of an isoparametric auxiliary operator, we first prove that the bilinear form is continuous with respect to the DG norm when its first argument is the isoparametric projection error. Then the geometric approximation error of inflow boundary of original domain is precisely estimated. The error order between discrete normal vectors and the continuous ones are also proven. Finally, the rigorous analysis yields an optimal convergence rate in the DG norm. Two- and three-dimensional numerical tests are conducted to support the theoretical results.