The <i>hp</i>‐multigrid method applied to <i>hp</i>‐adaptive refinement of triangular grids

Mitchell, William F.

doi:10.1002/nla.700

Cited by 22 publications

(15 citation statements)

References 48 publications

(46 reference statements)

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“…Similar approaches have been applied successfully in the context of hp-FEM by e.g. [2,81,82]. An alternative idea is to exploit the local nature of the overlay shape functions.…”

Section: Discussionmentioning

confidence: 97%

Multi-level hp-adaptivity: high-order mesh adaptivity without the difficulties of constraining hanging nodes

et al. 2015

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The implementation of hp-adaptivity is challenging as hanging nodes, edges, and faces have to be constrained to ensure compatibility of the shape functions. For this reason, most hp-code frameworks restrict themselves to 1-irregular meshes to ease the implementational effort. This work alleviates these difficulties by introducing a new formulation for high-order mesh adaptivity that provides full local hp-refinement capabilities at a comparably small implementational effort. Its main idea is the extension of the hp-d-method such that it allows for high-order overlay meshes yielding a hierarchical, multi-level hp-formulation of the Finite Element Method. This concept enables intuitive refinement and coarsening procedures, while linear independence and compatibility of the shape functions are guaranteed by construction. The proposed method is demonstrated to achieve exponential rates of convergence-both in terms of degrees of freedom and in run-time-for problems with nonsmooth solutions. Furthermore, the scheme is used alongside the Finite Cell Method to simulate the heat flow around moving objects on a non-conforming background mesh and is combined with an energy-based refinement indicator for automatic hp-adaptivity.

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“…Similar approaches have been applied successfully in the context of hp-FEM by e.g. [2,81,82]. An alternative idea is to exploit the local nature of the overlay shape functions.…”

Section: Discussionmentioning

confidence: 97%

Multi-level hp-adaptivity: high-order mesh adaptivity without the difficulties of constraining hanging nodes

et al. 2015

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“…In the present paper, we propose to use an a posteriori residual as reference for stopping MG cycles.Multigrid and FMG have been combined with mesh adaptation in many works. Let us cite a pioneering work of R. Bank [9] and a few more recent ones, such as [10][11][12][13]. Adaptative works are most frequently based on mesh refinement by local division, producing embedded meshes.…”

mentioning

confidence: 99%

A mesh‐adaptative metric‐based full multigrid for the Poisson problem

Brèthes

Allain

Dervieux

2015

Numerical Methods in Fluids

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Summary This paper studies the combination of the full‐multigrid (FMG) algorithm with an anisotropic metric‐based mesh adaptation algorithm. For the sake of simplicity, the case of an elliptic two‐dimensional partial differential equation is studied. Meshes are unstructured and non‐embedded, defined through the metric‐based parameterization. A rather classical MG preconditioner is applied, in combination with a quasi‐Newton fixed point. An anisotropic metric‐based mesh adaptation loop is introduced inside the FMG algorithm. FMG convergence stopping test is revisited. Applications to a few two‐dimensional continuous and discontinuous coefficient elliptic model problems show the efficiency of this combination. Copyright © 2015 John Wiley & Sons, Ltd.

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“…Nowadays, static condensation is frequently used to eliminate the geometrically interior degrees of freedom in high-order finite element discretizations, see e.g., [36, §3.5.9] [27]. A critical property of static condensation is that the reduced matrix S has the same sparsity pattern as the principal sub-matrix A rr , and therefore the solution of (1.4) always requires less memory than that of (1.1).…”

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confidence: 99%

“…Schur complement reduction, especially in the form of static condensation, has been applied in many areas of scientific computing, including mimetic discretizations for diffusion problems on generalized polyhedral meshes [8], hp-adaptive refinement for elliptic equations [27], discontinuous Galerkin methods [17] and physics-based preconditioning in magnetohydrodynamics (MHD) [15]. In a number of cases, AMG was used for solving the condensed system, leading to the central problem of interest in this paper.…”

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confidence: 99%

Algebraic Multigrid for Linear Systems Obtained by Explicit Element Reduction

Brunner¹,

Kolev²

2011

SIAM J. Sci. Comput.

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Abstract. We consider sparse linear systems, where a set of "interior" degrees of freedom have been eliminated in order to reduce the problem size. This elimination is assumed to be local, so the "interior" principal sub-matrix is block-diagonal, and the resulting Schur complement is still sparse. For it to be beneficial, the elimination process should lead to reduced memory requirements, but equally importantly, it should also result in an algebraic problem that can be solved efficiently. In this paper we propose a general element reduction approach, and show how the elimination process can exploit a particular "sub-zonal" discretization to maintain the sparsity of the Schur complement. We also investigate Algebraic Multigrid (AMG) solution algorithms applied to the reduced problem, and we discuss the influence of the local elimination on solver-related properties of the matrix, such as near-nullspace preservation and the availability of stable subspace decompositions. We focus on BoomerAMG, a parallel variant of classical Ruge-Stüben AMG, applied to scalar diffusion problems [18], and the Auxiliary-space Maxwell Solver (AMS) for electromagnetic diffusion applications [22]. In the electromagnetic case, we establish algebraically a reduced version of the HX decomposition from [19], and consider a modification of the reduction process that targets the singular problems arising in simulations with pure void (zero conductivity) regions. For scalar diffusion problems, our particular stencil analysis shows that the reduction has a positive effect on meshes with stretched elements. We present a number of 2D, 3D and axi-symmetric numerical experiments, which demonstrate that the combination of an appropriately chosen local elimination with the use of the BoomerAMG and AMS solvers can lead to significant improvements in the overall solution time.

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

Cited by 22 publications

References 48 publications

Multi-level hp-adaptivity: high-order mesh adaptivity without the difficulties of constraining hanging nodes

Multi-level hp-adaptivity: high-order mesh adaptivity without the difficulties of constraining hanging nodes

A mesh‐adaptative metric‐based full multigrid for the Poisson problem

Algebraic Multigrid for Linear Systems Obtained by Explicit Element Reduction

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