A fast spectral element solver combining static condensation and multigrid techniques

Lecture Notes in Computational Science and Engineering

2017

Abstract. We present a polynomial multigrid method for the nodal interior penalty formulation of the Poisson equation on three-dimensional Cartesian grids. Its key ingredient is a weighted overlapping Schwarz smoother operating on element-centered subdomains. The MG method reaches superior convergence rates corresponding to residual reductions of about two orders of magnitude within a single V(1,1) cycle. It is robust with respect to the mesh size and the ansatz order, at least up to P = 32. Rigorous exploitation of tensor-product factorization yields a computational complexity of O(P N ) for N unknowns, whereas numerical experiments indicate even linear runtime scaling. Moreover, by allowing adjustable subdomain overlaps and adding Krylov acceleration, the method proved feasible for anisotropic grids with element aspect ratios up to 48.

Section: Multigrid and Preconditioned Conjugate Gradient Methodsmentioning

confidence: 99%

Robust Multigrid for Cartesian Interior Penalty DG Formulations of the Poisson Equation in 3D

Lecture Notes in Computational Science and Engineering

2017

“…Current methods do not hold up to this standard. Multigrid with overlapping block-smoothers [1,2] is a good candidate, as it generates a constant iteration count, but it is not robust against the aspect ratio and the operators scale super-linearly when increasing the polynomial degree. The first problem can be remedied by introducing weighting functions, which works for SEM [3] as well as DG [4], however the scaling still remains.…”

Section: Introductionmentioning

confidence: 99%

Building blocks for a leading edge high‐order flow solver

Huismann

Proc Appl Math and Mech

Fröhlich

2017

An important trend in Computational Fluid Dynamics is towards high-order methods, as they offer a substantially lower discretization error for the same number of degrees of freedom (DOF). Examples are the Spectral-Element Methods (SEM) and Discontinuous Galerkin (DG) methods. Unfortunately, with most implementations the work load of such solvers increases drastically with the number of DOF, for example, when increasing the polynomial degree of the approximation. This issue gets particular pressing for elliptic solvers which are a vital building block in the time-stepping of the incompressible NavierStokes equations, resulting from pressure projection methods or implicit treatment of viscous terms. So far, this drastic increase of resources has hampered the use of SEM for higher polynomial degrees, such as 16 or more.The present contribution is located at this particular "Frontier of CFD" and proposes an SEM with linear scaling in the number of degrees of freedom. It is achieved independent of the polynomial degree, independent of the aspect ratio of elements, and involves constant iteration count when increasing the number of elements. The method is based on combining static condensation with block-Jacobi preconditioning and iterative substructuring. The latter two lead to a constant iteration count, while the former warrants linear operator complexity.In the presentation, the scheme is described in detail and applied to the construction of a Helmholtz solver. This solver is extremely fast and able to solve the Helmholtz equation with 10 10 unknowns on 240 cores in acceptable time. It thus enables competitive high-order SEM simulations even on small clusters and in this way expands the frontier of CFD towards highly accurate results requiring comparatively modest resources.

“…As a step into that direction we present a new p-multigrid method for interior penalty and local discontinuous Galerkin discretizations of the Poisson equation on Cartesian grids. Our approach is motivated and strongly influenced by previous work dedicated to the continuous spectral element method [16,20,26,33]. We propose two classes of multiplicative and weighted additive Schwarz methods, which use an adjustable overlap depending on the polynomial level.…”

Section: Introductionmentioning

confidence: 99%

Robust multigrid for high-order discontinuous Galerkin methods: A fast Poisson solver suitable for high-aspect ratio Cartesian grids

Journal of Computational Physics

2016

Abstract. We present a polynomial multigrid method for nodal interior penalty and local discontinuous Galerkin formulations of the Poisson equation on Cartesian grids. For smoothing we propose two classes of overlapping Schwarz methods. The first class comprises element-centered and the second face-centered methods. Within both classes we identify methods that achieve superior convergence rates, prove robust with respect to the mesh spacing and the polynomial order, at least up to P = 32. Consequent structure exploitation yields a computational complexity of O(P N ), where N is the number of unknowns. Further we demonstrate the suitability of the face-centered method for element aspect ratios up to 32.