Evaluation of Curved Element Discontinuous Galerkin Meshes

Abstract

The verification techniques developed for the method of manufactured solutions are utilized to study the effects mesh curvature on a high-order-accurate discontinuous Galerkin solver. Exact and nearby solutions are generated for domains with smoothly curving boundary surfaces. The accuracy of the solver is examined when the domain is resolved by linear, quadratic, cubic and Hermite element shapes. These observations are in agreement with other similar results reported in the literature. Finally, a newly developed test case is described which will enable verification data to be acquired in the regime of very highReynolds number flows near curved boundaries. I. Introduction In the past few years, there has been increased interest in using the discontinuous Galerkin method to solve the Euler, Navier-Stokes, and Reynolds-averaged Navier-Stokes equations. The favorable properties of the method have been well established in the literature, and numerous instances of its application to these problems have been documented. At present, there is considerable research being devoted to the method. Most of this effort is focused on developing algorithms to improve its overall computational efficiency, while still others are looking to extend the method for use into other application areas where high-order accuracy may be used to good advantage. The motivation for the present work stems from an effort to verify the correct implementation of a recently developed discontinuous Galerkin solver for the compressible Navier-Stokes equations. As with many other researchers before us, we have found that the solver is very sensitive to the mesh approximation of curved boundaries with linear elements. A search of the literature revealed numerous instances where such behavior has been observed and easily overcome by introducing curved mesh elements near the boundaries. The conventional wisdom in the finite element community is that isoparametric mesh elements (using the same order polynomials to approximate both the element and solution shapes) are required to maintain the formal accuracy of the solver. 1 Bassi and Rebay were among the first to perform a serious evaluation of the problem as it pertains to the use of the discontinuous Galerkin method to simulate inviscid fluid flows around curved boundaries. 2 @(�)