Numerical study on the effect of geometric approximation error in the numerical solution of PDEs using a high-order curvilinear mesh

TL;DR

This study investigates how geometric approximation errors in high-order curvilinear meshes affect the accuracy and stability of numerical solutions to PDEs on curved surfaces, using a specialized mesh generator to minimize such errors.

Contribution

It introduces the adaptation of NekMesh for generating high-order meshes with negligible geometric approximation error, enabling detailed analysis of geometric errors' impact on PDE solutions.

Findings

High-order meshes with minimal geometric error improve PDE solution accuracy.

Geometric approximation errors significantly affect conservation properties in numerical schemes.

The impact varies with polynomial order and PDE type.

Abstract

When time-dependent partial differential equations (PDEs) are solved numerically in a domain with curved boundary or on a curved surface, mesh error and geometric approximation error caused by the inaccurate location of vertices and other interior grid points, respectively, could be the main source of the inaccuracy and instability of the numerical solutions of PDEs. The role of these geometric errors in deteriorating the stability and particularly the conservation properties are largely unknown, which seems to necessitate very fine meshes especially to remove geometric approximation error. This paper aims to investigate the effect of geometric approximation error by using a high-order mesh with negligible geometric approximation error, even for high order polynomial of order p. To achieve this goal, the high-order mesh generator from CAD geometry called NekMesh is adapted for surface mesh generation in comparison to traditional meshes with non-negligible geometric approximation error. Two types of numerical tests are considered. Firstly, the accuracy of differential operators is compared for various p on a curved element of the sphere. Secondly, by applying the method of moving frames, four different time-dependent PDEs on the sphere are numerically solved to investigate the impact of geometric approximation error on the accuracy and conservation properties of high-order numerical schemes for PDEs on the sphere.