Local mesh refinement in 2 and 3 dimensions

Bänsch, Eberhard

doi:10.1016/0899-8248(91)90006-g

Cited by 289 publications

(183 citation statements)

References 6 publications

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“…The algorithm terminates if none of these are left for refinement. Bänsch (1991) proves that this algorithm always terminates and leads to a conforming triangulation. Coarsening is also possible.…”

Section: (B) Geometry Of Local Mesh Refinementmentioning

confidence: 99%

Efficiency considerations in triangular adaptive mesh refinement

Behrens

Bäder

2009

Phil. Trans. R. Soc. A.

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Locally or adaptively refined meshes have been successfully applied to simulation applications involving multi-scale phenomena in the geosciences. In particular, for situations with complex geometries or domain boundaries, meshes with triangular or tetrahedral cells demonstrate their superior ability to accurately represent relevant realistic features.On the other hand, these methods require more complex data structures and are therefore less easily implemented, maintained and optimized. Acceptance in the Earthsystem modelling community is still low. One of the major drawbacks is posed by indirect addressing due to unstructured or dynamically changing data structures and correspondingly lower efficiency of the related computations.In this paper, we will derive several strategies to circumvent the mentioned efficiency constraint. In particular, we will apply recent computational sciences methods in combination with results of classical mathematics (space-filling curves) in order to linearize the complex data and access structure.

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Section: (B) Geometry Of Local Mesh Refinementmentioning

confidence: 99%

Efficiency considerations in triangular adaptive mesh refinement

Behrens

Bäder

2009

Phil. Trans. R. Soc. A.

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“…There are several bisection methods proposed for d ≥ 3 [8,32,33,46,3,54], which generalize the newest vertex bisection [37] and longest edge bisection [48] for d = 2. We now give a mathematical description based on Kossaczky [32], Traxler [55], and Stevenson [54].…”

Section: Bisection Rulesmentioning

confidence: 99%

“…The module REFINE refines all marked elements and perhaps a few more to keep mesh conformity. Of all possible refinement strategies, we are interested in bisection, a popular, elegant, and effective procedure for refinement in any dimension [48,8,32,33,55,3,49,54]. Our goal is to design optimal multilevel solvers that constitute the core of procedure SOLVE, and analyze them within the framework of highly graded meshes created by bisection, from now on called bisection grids.…”

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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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“…Each marked triangle is refined once with the help of the bisection algorithm that is described in Bänsch [Bän91a,Bän91b]. In this way the shape-regularities of refined triangulations depend only on the shape-regularity of the initial triangulation.…”

Section: A Posteriori Analysis For Prescribed Mean Curvature Equationmentioning

confidence: 99%

On the a posteriori error analysis for equations of prescribed mean curvature

Fierro

Veeser

2003

Math. Comp.

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Abstract. We present two approaches to the a posteriori error analysis for prescribed mean curvature equations. The main difference between them concerns the estimation of the residual: without or with computable weights. In the second case, the weights are related to the eigenvalues of the underlying operator and thus provide local and computable information about the conditioning. We analyze the two approaches from a theoretical viewpoint. Moreover, we investigate and compare the performance of the derived indicators in an adaptive procedure. Our theoretical and practical results show that it is advantageous to estimate the residual in a weighted way.

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Local mesh refinement in 2 and 3 dimensions

Cited by 289 publications

References 6 publications

Efficiency considerations in triangular adaptive mesh refinement

Efficiency considerations in triangular adaptive mesh refinement

Optimal multilevel methods for graded bisection grids

On the a posteriori error analysis for equations of prescribed mean curvature

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