Analysis of ``p''-Multigrid for Continuous and Discontinuous Finite Element Discretizations

Helenbrook, Brian T.; Mavriplis, Dimitri J.; Atkins, Harold L.

doi:10.2514/6.2003-3989

Cited by 87 publications

(60 citation statements)

References 6 publications

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“…Compared to the 1D results shown in table 5, these results are all worse. This behavior also occurs when using p-multigrid to solve continuous spectral element formulations [9]. The explanation for this behavior that is usually given is that a high-order finite-element simulation corresponds to a discretization that is on a high aspect ratio mesh; the spacing of Gauss Legendre points near ξ = −1 or 1 goes like 1/p 2 as compared to 1/p if the spacing is uniform.…”

Section: D Resultsmentioning

confidence: 98%

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Application of p-Multigrid to Discontinuous Galerkin Formulations of the Poisson Equation

Helenbrook

Atkins

2006

AIAA Journal

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We investigate p-multigrid as a solution method for several different discontinuous Galerkin (DG) formulations of the Poisson equation. Different combinations of relaxation schemes and basis sets have been combined with the DG formulations to find the best performing combination. The damping factors of the schemes have been determined using Fourier analysis for both one and two-dimensional problems. One important finding is that when using DG formulations, the standard approach of forming the coarse p matrices separately for each level of multigrid is often unstable. To ensure stability the coarse p matrices must be constructed from the fine grid matrices using algebraic multigrid techniques. Of the relaxation schemes, we find that the combination of Jacobi relaxation with the spectral element basis is fairly effective. The results using this combination are p sensitive in both one and two dimensions, but reasonable convergence rates can still be achieved for moderate values of p and isotropic meshes. A competitive alternative is a block Gauss-Seidel relaxation. This actually out performs a more expensive line relaxation when the mesh is isotropic. When the mesh becomes highly anisotropic, the implicit line method and the Gauss-Seidel implicit line method are the only effective schemes. Adding the Gauss-Seidel terms to the implicit line method gives a significant improvement over the line relaxation method.

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Section: D Resultsmentioning

confidence: 98%

“…For discontinuous formulations, we have performed some analysis of p-multigrid for hyperbolic systems [9], but not for elliptic equations.…”

Section: Introductionmentioning

confidence: 99%

Application of p-Multigrid to Discontinuous Galerkin Formulations of the Poisson Equation

Helenbrook

Atkins

2006

AIAA Journal

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“…For the first time, a geometric multigrid method was used for the solution at p = 1. Two years later, Helenbrook et al [25] changed the name of the method to ' p-multigrid', but still used a geometric p sequence. This paper considered both a (non-adaptive) hp-FEM discretization of the Laplace equation in 2D, and Streamline-Upwind Petrov-Galerkin (SUPG) and discontinuous Galerkin discretizations of the convection equation.…”

Section: Historymentioning

confidence: 99%

The hp‐multigrid method applied to hp‐adaptive refinement of triangular grids

Mitchell

2010

Numerical Linear Algebra App

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SUMMARYRecently the hp version of the finite element method, in which adaptivity occurs in both the size, h, of the elements and in the order, p, of the approximating piecewise polynomials, has received increasing attention. It is desirable to combine this optimal order discretization method with an optimal order algebraic solution method like multigrid. An intriguing notion is to use the values of p as the levels of a multilevel method. In this paper we present such a method, known as hp-multigrid, for high-order finite elements and hp-adaptive grids. We present a survey of the development of p-multigrid and hp-multigrid, define an hp-multigrid algorithm based on the p-hierarchical basis for the p levels and h-hierarchical basis for an h-multigrid solution of the p = 1 'coarse grid' equations, and present numerical convergence results using hp-adaptive grids. The numerical results suggest that the method has a convergence rate of

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“…[5][6][7]12 Efforts to develop efficient implicit solvers for DG have included optimized relaxation schemes, 11 analysis of traditional geometric multigrid, 13 GMRES, [14][15][16] and P-multigrid. [17][18][19][20] Analysis of geometric multigrid indicates mesh that independent results are possible; however, the required restriction operators are complex and would be difficult to implement for general unstructured grids. A large effort in GMRES can be found in open literature; however the method incurs a large storage penalty, and none have achieved mesh independent convergence for high Reynolds number flows (or other highly stiff cases).…”

Section: 10mentioning

confidence: 99%

Numerical Evaluation of P-Multigrid Method for the Solution of Discontinuous Galerkin Discretizations of Diffusive Equations

Atkins

Helenbrook

2005

17th AIAA Computational Fluid Dynamics Conference

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This paper describes numerical experiments with P-multigrid to corroborate analysis, validate the present implementation, and to examine issues that arise in the implementations of the various combinations of relaxation schemes, discretizations and P-multigrid methods. The two approaches to implement P-multigrid presented here are equivalent for most high-order discretization methods such as spectral element, SUPG, and discontinuous Galerkin applied to advection; however it is discovered that the approach that mimics the common geometric multigrid implementation is less robust, and frequently unstable when applied to discontinuous Galerkin discretizations of diffusion. Gauss-Seidel relaxation converges ≈ 40% faster than block Jacobi, as predicted by analysis; however, the implementation of Gauss-Seidel is considerably more expensive that one would expect because gradients in most neighboring elements must be updated. A compromise quasi Gauss-Seidel relaxation method that evaluates the gradient in each element twice per iteration converges at rates similar to those predicted for true Gauss-Seidel.

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Analysis of ``p''-Multigrid for Continuous and Discontinuous Finite Element Discretizations

Cited by 87 publications

References 6 publications

Application of p-Multigrid to Discontinuous Galerkin Formulations of the Poisson Equation

Application of p-Multigrid to Discontinuous Galerkin Formulations of the Poisson Equation

The hp‐multigrid method applied to hp‐adaptive refinement of triangular grids

Numerical Evaluation of P-Multigrid Method for the Solution of Discontinuous Galerkin Discretizations of Diffusive Equations

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