Error Estimates for Adaptive Finite Element Computations

Babuvška, I.; Rheinboldt, Werner C.

doi:10.1137/0715049

Cited by 1,266 publications

(618 citation statements)

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“…This error estimator essentially consists of the elementwise error of the finite element functions with respect to the strong form of the differential equation and of jumps across inter-element boundaries of that boundary operator which naturally links the strong and weak forms of the differential equation. Some of the results of § §6-8 are completely new, others are generalizations of, and improvements upon, results previously obtained in [4,5,7,8,9,19,25,27,28]. We then trivially have a-l(\\F(u)\\Y-) < ||" -"oik < Q-x(\\F(u)\\Y.)…”

Section: Introductionmentioning

confidence: 78%

“…We conclude this section with a simple example of an a posteriori error estimator which is based on the solution of auxiliary local problems and which generalizes the estimator introduced in [4,5]. For simplicity we assume that p = q = r = 2.…”

Section: Estimation Of the Residualmentioning

confidence: 99%

“…D When combining Propositions 2.1 and 4.1 we obtain a residual a posteriori error estimator. The following proposition together with Proposition 2.1 yields a framework for some of those a posteriori error estimators which are based on the solution of auxiliary local problems, such as the one described in [4,5]. Let ^,, h > 0, be a family of partitions of Í2 into «-simplices, which satisfies the following conditions:…”

Section: Estimation Of the Residualmentioning

confidence: 99%

“…In this section we also give a framework which covers some of the a posteriori error estimators based on the solution of auxiliary local problems, such as the one described in [4,5], and which shows that these estimators are equivalent to the residual a posteriori error estimator considered before.…”

Section: Introductionmentioning

confidence: 99%

“…(4) averaging methods: Use some local averaging technique for error estimation (cf., e.g., [6,21,29,30]). …”

Section: Introductionmentioning

confidence: 99%

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A posteriori error estimates for nonlinear problems. Finite element discretizations of elliptic equations

Verfürth¹

1994

Math. Comp.

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Abstract. We give a general framework for deriving a posteriori error estimates for approximate solutions of nonlinear problems. In a first step it is proven that the error of the approximate solution can be bounded from above and from below by an appropriate norm of its residual. In a second step this norm of the residual is bounded from above and from below by a similar norm of a suitable finite-dimensional approximation of the residual. This quantity can easily be evaluated, and for many practical applications sharp explicit upper and lower bounds are readily obtained. The general results are then applied to finite element discretizations of scalar quasi-linear elliptic partial differential equations of 2nd order, the eigenvalue problem for scalar linear elliptic operators of 2nd order, and the stationary incompressible Navier-Stokes equations. They immediately yield a posteriori error estimates, which can easily be computed from the given data of the problem and the computed numerical solution and which give global upper and local lower bounds on the error of the numerical solution.

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Section: Introductionmentioning

confidence: 78%

Section: Estimation Of the Residualmentioning

confidence: 99%

Section: Estimation Of the Residualmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…(4) averaging methods: Use some local averaging technique for error estimation (cf., e.g., [6,21,29,30]). …”

Section: Introductionmentioning

confidence: 99%

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A posteriori error estimates for nonlinear problems. Finite element discretizations of elliptic equations

Verfürth¹

1994

Math. Comp.

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A 3-D Adaptive Mesh Refinement Using Variable-Node Solid Transition Elements

Choi

Lee

1996

Int. J. Numer. Meth. Engng.

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SUMMARYAn automated three-dimensional adaptive h-refinement strategy using the solid transition elements with variable midside nodes at edges and faces of the element is presented. The basic behaviour of these transition elements were improved by addition of associated non-conforming modes. By introducing these transition elements, some difficulties associated with imposing displacement constraints on irregular nodes to enforce interelement compatibility in the conventional adaptive h-refinement are easily overcome. A superconvergent patch recovery technique is also extended to three-dimensional problem. Numerical examples show the effectiveness of the proposed adaptive mesh refinement scheme using transition elements.

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