Optimal Multilevel Iterative Methods for Adaptive Grids

Mitchell, William F.

doi:10.1137/0913009

Cited by 80 publications

(75 citation statements)

References 23 publications

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“…m ≥ 2, one can include all degree of freedom (dof) inside the bisection patch ω i into the subspaceṼ i and obtain a stable decomposition. The corresponding local multigrid methods then requires smoothing on all dof inside ω i ; see [39].…”

Section: Generation Of Compatiblementioning

confidence: 99%

Optimal multilevel methods for graded bisection grids

Chen

Nochetto

2011

Numer. Math.

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Abstract. We design and analyze optimal additive and multiplicative multilevel methods for solving H 1 problems on graded grids obtained by bisection. We deal with economical local smoothers: after a global smoothing in the finest mesh, local smoothing for each added node during the refinement needs to be performed only for three vertices -the new vertex and its two parent vertices. We show that our methods lead to optimal complexity for any dimensions and polynomial degree. The theory hinges on a new decomposition of bisection grids in any dimension, which is of independent interest and yields a corresponding decomposition of spaces. We use the latter to bridge the gap between graded and quasi-uniform grids, for which the multilevel theory is well-established.

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Section: Generation Of Compatiblementioning

confidence: 99%

Optimal multilevel methods for graded bisection grids

Chen

Nochetto

2011

Numer. Math.

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“…If we use the bisection algorithm (cf. e.g., [22,24]) for mesh refinements, l is the set of new vertices and their immediately neighboring vertices (cf. [32]) and l is the set of new edges and their immediately neighboring edges (see Fig.…”

Section: Multigrid V-cycle Algorithms Inmentioning

confidence: 99%

“…To keep the optimal computational cost on locally refined meshes, one must adopt the local multigrid policy [3,22,32], which confines relaxations to degrees of freedom on new elements of each mesh level. Clearly this policy makes the computational cost of the local multigrid method proportional to the number of all elements appearing in the local refinement process, and thus proportional to the number of degrees of freedom on the finest mesh.…”

Section: Introductionmentioning

confidence: 99%

Uniform Convergence of Adaptive Multigrid Methods for Elliptic Problems and Maxwell's Equations

Hiptmair¹

2012

NMTMA

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Abstract.We consider the convergence theory of adaptive multigrid methods for secondorder elliptic problems and Maxwell's equations. The multigrid algorithm only performs pointwise Gauss-Seidel relaxations on new degrees of freedom and their "immediate" neighbors. In the context of lowest order conforming finite element approximations, we present a unified proof for the convergence of adaptive multigrid V-cycle algorithms. The theory applies to any hierarchical tetrahedral meshes with uniformly bounded shape-regularity measures. The convergence rates for both problems are uniform with respect to the number of mesh levels and the number of degrees of freedom. We demonstrate our convergence theory by two numerical experiments.

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“…Let T be a refinement of S which is defined as follows. According to a certain rule (such as that of [ 6 ] ) , some of the triangles in S are refined. If s E S is refined, then connect the midpoints of its edges to each other and include all the resulting triangles in 7'.…”

Section: Preliminariesmentioning

confidence: 99%

“…This refinement is illustrated for d = 2 in Figure 1 (other refinement strategies are also possible, see [a] [6]). …”

Section: Preliminariesmentioning

confidence: 99%

Multigrid for refined triangle meshes

Shapira¹

1997

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Optimal Multilevel Iterative Methods for Adaptive Grids

Cited by 80 publications

References 23 publications

Optimal multilevel methods for graded bisection grids

Optimal multilevel methods for graded bisection grids

Uniform Convergence of Adaptive Multigrid Methods for Elliptic Problems and Maxwell's Equations

Multigrid for refined triangle meshes

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