Adaptation strategies for high order discontinuous Galerkin methods based on Tau-estimation

Abstract

In this paper three p-adaptation strategies based on the minimization of the truncation error are presented for high order discontinuous Galerkin methods. The truncation error is approximated by means of a r-estimation procedure and enables the identification of mesh regions that require adaptation. Three adaptation strategies are developed and termed a posteriori, quasi-a priori and quasi-a priori corrected. All strategies require fine solutions, which are obtained by enriching the polynomial order, but while the former needs time converged solutions, the last two rely on non-converged solutions, which lead to faster computations. In addition, the high order method permits the spatial decoupling for the estimated errors and enables anisotropic p-adaptation. These strategies are verified and compared in terms of accuracy and computational cost for the Euler and the compressible Navier-Stokes equations. It is shown that the two quasia priori methods achieve a significant reduction in computational cost when compared to a uniform polynomial enrichment. Namely, for a viscous boundary layer flow, we obtain a speedup of 6.6 and 7.6 for the quasi-a priori and quasi-a priori corrected approaches, respectively.

Very high order discontinuous Galerkin method in elliptic problems

“…in the DG method [11,23,24,29,31,35,43] and in the finite element method [5,8,16,20,26,27]. This paper exploits the version the discontinuous Galerkin method where the continuity and boundary conditions are enforced by the finite difference rule.…”

Sensitivity Analysis to Control the Far-Wake Unsteadiness Behind Turbines

“…High order methods (order ≥ 3) are characterised by low numerical errors (i.e., dispersion and diffusion) and their ability to use mesh refinement (increased number of mesh nodes or h-refinement) and/or polynomial enrichment (p-refinement) to achieve more accurate solutions [46,47]. The latter p-refinement method provides an exponential decay of the numerical error rather than the typical algebraic decay yielded by h-refinement solvers.…”

“…The Mach number is selected low, M = 0.2, such that compressibility effects are negligible. Details of the numerical solver may be found in [46][47][48].…”

Adaptation Strategies for Discontinuous Galerkin Spectral Element Methods by means of Truncation Error Estimations

“…Having determined this trend, it is possible to use extrapolation to find higher polynomials that result in a particularly lower error and save computational cost at the same time [Kompenhans 2016]. …”