Truncation Error Estimation in the p-Anisotropic Discontinuous Galerkin Spectral Element Method

TL;DR

This paper introduces a new anisotropic truncation error estimator for DGSEM that improves accuracy and reduces computational cost in p-adaptation by exploiting tensor product basis properties.

Contribution

A novel anisotropic truncation error estimator for DGSEM based on $ au$-estimation, offering improved accuracy and efficiency for p-adaptation strategies.

Findings

The new estimator is computationally cheaper than previous methods.

It provides more accurate error extrapolations for higher polynomial orders.

Robustness allows using coarser reference solutions, reducing costs.

Abstract

In the context of Discontinuous Galerkin Spectral Element Methods (DGSEM), $\tau$-estimation has been successfully used for p-adaptation algorithms. This method estimates the truncation error of representations with different polynomial orders using the solution on a reference mesh of relatively high order. In this paper, we present a novel anisotropic truncation error estimator derived from the $\tau$-estimation procedure for DGSEM. We exploit the tensor product basis properties of the numerical solution to design a method where the total truncation error is calculated as a sum of its directional components. We show that the new error estimator is cheaper to evaluate than previous implementations of the $\tau$-estimation procedure and that it obtains more accurate extrapolations of the truncation error for representations of a higher order than the reference mesh. The robustness of the method allows performing the p-adaptation strategy with coarser reference solutions, thus further reducing the computational cost. The proposed estimator is validated using the method of manufactured solutions in a test case for the compressible Navier-Stokes equations.