Quasi-a priori mesh adaptation and extrapolation to higher order using τ-estimation

Journal of Computational Physics

et al. 2019

High-order discontinuous Galerkin methods have become a popular technique in computational fluid dynamics because their accuracy increases spectrally in smooth solutions with the order of the approximation. However, their main drawback is that increasing the order also increases the computational cost. Several techniques have been introduced in the past to reduce this cost. On the one hand, local mesh adaptation strategies based on error estimation have been proposed to reduce the number of degrees of freedom while keeping a similar accuracy. On the other hand, multigrid solvers may accelerate time marching computations for a fixed number of degrees of freedom.In this paper, we combine both methods and present a novel anisotropic p-adaptation multigrid algorithm for steady-state problems that uses the multigrid scheme both as a solver and as an anisotropic error estimator. To achieve this, we show that a recently developed anisotropic truncation error estimator [1, A. M. Rueda-Ramírez, G. Rubio, E. Ferrer, E. Valero, Truncation Error Estimation in the p-Anisotropic Discontinuous Galerkin Spectral Element Method, Journal of Scientific Computing] is perfectly suited to be performed inside the multigrid cycle with negligible extra cost. Furthermore, we introduce a multi-stage p-adaptation procedure which reduces the computational time when very accurate results are required.The proposed methods are tested for the compressible Navier-Stokes equations, where we investigate two cases. First, the 2D boundary layer flow on a flat plate is studied to assess accuracy and computational cost of the algorithm, where a speed-up of 816 is achieved compared to the traditional explicit method. Second, the 3D flow around a sphere is simulated and used to test the anisotropic properties of the proposed method, where a speed-up of 152 is achieved compared to the explicit method. The proposed multi-stage procedure achieved a speed-up of 2.6 in comparison to the single-stage method in highly accurate simulations.

Section: Single-stage Adaptationmentioning

confidence: 86%

Section: Anisotropic Multigridmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A p-multigrid strategy with anisotropic p-adaptation based on truncation errors for high-order discontinuous Galerkin methods

Rueda-Ramírez

Manzanero

Journal of Computational Physics

et al. 2019

“…In this work, we focus on truncation error estimates since it has been shown that a reduction of the truncation error controls the numerical accuracy of all functionals [10], hence reducing the truncation error necessarily leads to a more accurate lift and drag. The τ -estimation method [4] is a way to estimate the truncation error locally that has been used to drive mesh adaptation strategies in low-order [9,20] and high-order methods [10,17,18]. The adaptation strategy consists in converging a high order representation (reference mesh) to a specified global residual and then performing a single error estimation followed by a corresponding mesh adaptation process.…”

Section: Introductionmentioning

confidence: 99%

An Anisotropic p-Adaptation Multigrid Scheme for Discontinuous Galerkin Methods

Rueda-Ramírez

Rubio

Lecture Notes in Computational Science and Engineering

et al. 2020

In this work, we present an anisotropic p-adaptation multigrid algorithm for discontinuous Galerkin methods for steady-state problems that uses a p-multigrid scheme both as a solver and as an anisotropic error estimator. To achieve this, we develop a new anisotropic truncation error estimator based on the tau-estimation method, that can be evaluated inside the multigrid cycle with a negligible extra cost. The new error estimator is cheaper to evaluate and more accurate than previous versions of the tau-estimation procedure. In our technique, a non-converged solution in a reference mesh is used to estimate the truncation error with the multigrid scheme for different combinations of polynomial orders in different directions inside every element, and the mesh is adapted accordingly to target a desired truncation error threshold. The accuracy and computational cost of the proposed p-anisotropic adaptation algorithm is tested for the steady viscous flow past a NACA0012 airfoil. A speed-up of 16 can be achieved in the proposed numerical example compared with the uniformly refined simulation without multigrid.

“…This relationship provides a valid argument for the use of the truncation error as a sensor for a mesh adaptation algorithm, e.g. Syrakos et al [47], Frayssee et al [20,21]. Moreover, in high order discretizations, an accurate estimate for the error also enables modification of the polynomial order via a procedure known as r-extrapolation to better capture the numerical solution (see Bernert [9]).…”

Section: Introductionmentioning

confidence: 99%

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

Kompenhans

Rubio

Journal of Computational Physics

et al. 2016

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.