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

Abstract

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.

An Anisotropic p-Adaptation Multigrid Scheme for Discontinuous Galerkin Methods

“…It has been shown that the use of FAS p-multigrid methods speeds up the computation of steady-state and unsteady solutions of the compressible Navier-Stokes equations [2,8]. In addition, Rueda-Ramírez et al [18] showed that the truncation error of an anisotropic representation can be inexpensively obtained inside an anisotropic p-multigrid cycle that performs the coarsening in one coordinate direction at a time. In fact, the second term of Eq.…”

“…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.…”

“…In this work, we employ the anisotropic truncation error estimator developed in [19] and the anisotropic p-adaptation method detailed in [18] to accelerate the computation of the compressible steady viscous flow past a NACA0012 at angle of attack 5 • , Re ∞ = 200 based on the airfoil chord, and M ∞ = 0.2. This particular settings correspond to a steady laminar flow, but the proposed method can be directly used with any steady solution (e.g.…”

A wavelet‐based variational multiscale method for the LES of incompressible flows in a high‐order DG‐FEM framework

“…This simple test case serves to illustrate the potential of the elementwise SGWT as an error indicator for carrying out h ‐adaptivity within the framework of a DG‐FEM discretization and constitutes an interesting prospect for future research. A similar application of this procedure to a laminar incompressible test case has been demonstrated by Pinto et al We must point out that p ‐adaptivity is also well suited to be applied within the DG‐FEM framework, as indicated by several recent works . It is highly likely that the elementwise SGWT error indicator presented above may be appropriate for use within this setting as well.…”

“…A similar application of this procedure to a laminar incompressible test case has been demonstrated by Pinto et al 82 We must point out that p-adaptivity is also well suited to be applied within the DG-FEM framework, as indicated by several recent works. [83][84][85] It is highly likely that the elementwise SGWT error indicator presented above may be appropriate for use within this setting as well.…”