Theory of adaptive finite element methods: An introduction

Nochetto, Ricardo H.; Siebert, Kunibert G.; Veeser, Andreas

doi:10.1007/978-3-642-03413-8_12

Cited by 206 publications

(249 citation statements)

References 60 publications

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“…Stability of mixed-type FEMs for saddle-point problems is verified in terms of the Babuška-Brezzi inf-sup condition [8,26]. The finite elements are adapted to local variations of the solution: adaptivity is often tuned by a posteriori estimates [14,22,157].…”

Section: 4mentioning

confidence: 99%

A review of numerical methods for nonlinear partial differential equations

Tadmor

2012

Bull. Amer. Math. Soc.

142

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Abstract. Numerical methods were first put into use as an effective tool for solving partial differential equations (PDEs) by John von Neumann in the mid1940s. In a 1949 letter von Neumann wrote "the entire computing machine is merely one component of a greater whole, namely, of the unity formed by the computing machine, the mathematical problems that go with it, and the type of planning which is called by both." The "greater whole" is viewed today as scientific computation: over the past sixty years, scientific computation has emerged as the most versatile tool to complement theory and experiments, and numerical methods for solving PDEs are at the heart of many of today's advanced scientific computations. Numerical solutions found their way from financial models on Wall Street to traffic models on Main Street. Here we provide a bird's eye view on the development of these numerical methods with a particular emphasis on nonlinear PDEs.

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Section: 4mentioning

confidence: 99%

A review of numerical methods for nonlinear partial differential equations

Tadmor

2012

Bull. Amer. Math. Soc.

142

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“…Consequently, if v = J k=0 v k is a stable decomposition satisfying (2.10), we get [53,42], and present a novel decomposition of conforming meshes obtained by bisection. We do not discuss the alternative refinement method, called regular refinement, which divides one simplex into 2 d children; see [7,24] …”

Section: We Deducementioning

confidence: 99%

“…This leads to a fixed number of similarity classes of elements, depending only on T 0 , and thus B1 holds for any d. We refer to [42] for a thorough discussion. We recall that for d = 2, rule (3.1) reduces to the newest vertex bisection, in which case Sewell [51] showed that all the descendants of a triangle in T 0 fall into four similarity classes and hence (B1) holds.…”

Section: Bisection Rulesmentioning

confidence: 99%

“…The algorithms proposed by Kossaczký [32] for d = 3 and Stevenson [54] for d ≥ 3 enforce such initial labeling upon further refining every element of the initial triangulation, which deteriorates the shape regularity. Although (B2) imposes a severe restriction on the initial labeling, we emphasize that it is also used to prove the optimal cardinality of adaptive finite element methods [53,18,42]. Finding conditions weaker than (B2) is a challenging open problem.…”

Section: Bisection Rulesmentioning

confidence: 99%

“…The most standard AFEM has been later examined by Cascón, Kreuzer, Nochetto and Siebert [18], who have proved a contraction property and quasi-optimal cardinality. We refer to Nochetto, Siebert and Veeser [42] for an introduction to the theory of adaptive finite element methods. The gap to obtaining optimal complexity is precisely having optimal solvers and storage for adaptive bisection grids -the topic of this paper.…”

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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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Approximation Theory and Methods

2011

Green's Functions and Boundary Value Problems

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Theory of adaptive finite element methods: An introduction

Cited by 206 publications

References 60 publications

A review of numerical methods for nonlinear partial differential equations

A review of numerical methods for nonlinear partial differential equations

Optimal multilevel methods for graded bisection grids

Approximation Theory and Methods

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