Error Estimates for Adaptive Finite Element Computations
01 Aug 1978-SIAM Journal on Numerical Analysis (Society for Industrial and Applied Mathematics)-Vol. 15, Iss: 4, pp 736-754
TL;DR: The main theorem gives an error estimate in terms of localized quantities which can be computed approximately, and the estimate is optimal in the sense that, up to multiplicative constants which are independent of the mesh and solution, the upper and lower error bounds are the same.
Abstract: A mathematical theory is developed for a class of a-posteriors error estimates of finite element solutions. It is based on a general formulation of the finite element method in terms of certain bilinear forms on suitable Hilbert spaces. The main theorem gives an error estimate in terms of localized quantities which can be computed approximately. The estimate is optimal in the sense that, up to multiplicative constants which are independent of the mesh and solution, the upper and lower error bounds are the same. The theoretical results also lead to a heuristic characterization of optimal meshes, which in turn suggests a strategy for adaptive mesh refinement. Some numerical examples show the approach to be very effective.
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Book•
01 Jan 2000TL;DR: In this paper, a summary account of the subject of a posteriori error estimation for finite element approximations of problems in mechanics is presented, focusing on methods for linear elliptic boundary value problems.
Abstract: This monograph presents a summary account of the subject of a posteriori error estimation for finite element approximations of problems in mechanics. The study primarily focuses on methods for linear elliptic boundary value problems. However, error estimation for unsymmetrical systems, nonlinear problems, including the Navier-Stokes equations, and indefinite problems, such as represented by the Stokes problem are included. The main thrust is to obtain error estimators for the error measured in the energy norm, but techniques for other norms are also discussed.
2,607 citations
Cites methods from "Error Estimates for Adaptive Finite..."
...During the period 1978-1983, a number of results for explicit error estimation techniques were obtained: We mention references [20] and (21] as representative of the work....
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...The estimator on element K is 77K = lIGx(ux) - 4AlL2(K)~ (2.61) The estimator is precisely the estimator originally proposed and analyzed by Babuska and Rheinboldt [12](Definition 6.3)....
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...The constraint may be relaxed by making use of a device used by Babuska and Rheinboldt [12]....
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...The subdomain residual method was devised by Babu§ka and Rheinboldt [20, 22, 23]....
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...The subdomain residual method A method for a posteriori error estimation based on solving local residual problems with homogeneous essential boundary data over small patches or subdomains of the domain 0 was devised by Babuska and Rheinboldt [13]....
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Book•
28 May 1996
TL;DR: Introduction.
Abstract: Introduction. A Simple Model Problem. Abstract Nonlinear Equations. Finite Element Discretizations of Elliptic PDEs. Practical Implementation. Bibliography. Subject Index.
2,253 citations
Book•
01 Jan 1982TL;DR: This work presents an adaptive method based on the idea of multiple, component grids for the solution of hyperbolic partial differential equations using finite difference techniques based upon Richardson-type estimates of the truncation error, which is a mesh refinement algorithm in time and space.
Abstract: We present an adaptive method based on the idea of multiple, component grids for the solution of hyperbolic partial differential equations using finite difference techniques. Based upon Richardson-type estimates of the truncation error, refined grids are created or existing ones removed to attain a given accuracy for a minimum amount of work. Our approach is recursive in that fine grids can themselves contain even finer grids. The grids with finer mesh width in space also have a smaller mesh width in time, making this a mesh refinement algorithm in time and space. We present the algorithm, data structures and grid generation procedure, and conclude with numerical examples in one and two space dimensions.
2,120 citations
TL;DR: The ‘dual-weighted-residual method’ is introduced initially within an abstract functional analytic setting, and is then developed in detail for several model situations featuring the characteristic properties of elliptic, parabolic and hyperbolic problems.
Abstract: This article surveys a general approach to error control and adaptive mesh design in Galerkin finite element methods that is based on duality principles as used in optimal control. Most of the existing work on a posteriori error analysis deals with error estimation in global norms like the ‘energy norm’ or the L2 norm, involving usually unknown ‘stability constants’. However, in most applications, the error in a global norm does not provide useful bounds for the errors in the quantities of real physical interest. Further, their sensitivity to local error sources is not properly represented by global stability constants. These deficiencies are overcome by employing duality techniques, as is common in a priori error analysis of finite element methods, and replacing the global stability constants by computationally obtained local sensitivity factors. Combining this with Galerkin orthogonality, a posteriori estimates can be derived directly for the error in the target quantity. In these estimates local residuals of the computed solution are multiplied by weights which measure the dependence of the error on the local residuals. Those, in turn, can be controlled by locally refining or coarsening the computational mesh. The weights are obtained by approximately solving a linear adjoint problem. The resulting a posteriori error estimates provide the basis of a feedback process for successively constructing economical meshes and corresponding error bounds tailored to the particular goal of the computation. This approach, called the ‘dual-weighted-residual method’, is introduced initially within an abstract functional analytic setting, and is then developed in detail for several model situations featuring the characteristic properties of elliptic, parabolic and hyperbolic problems. After having discussed the basic properties of duality-based adaptivity, we demonstrate the potential of this approach by presenting a selection of results obtained for practical test cases. These include problems from viscous fluid flow, chemically reactive flow, elasto-plasticity, radiative transfer, and optimal control. Throughout the paper, open theoretical and practical problems are stated together with references to the relevant literature.
1,274 citations
Cites methods from "Error Estimates for Adaptive Finite..."
...As our quality measure, we use the over-estimation factor Ieff := ∣∣∣η(uh) J(e) ∣∣∣, referred to as the (reciprocal) ‘effectivity index’ (see Babuška and Rheinboldt (1978b))....
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...This approach was initiated by the pioneering work of Babuška and Rheinboldt (1978a, 1978b) and was then further developed by Ladeveze and Leguillon (1983), Bank and Weiser (1985), and Babuška and Miller (1987), to mention only a few of the most influential papers....
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TL;DR: In this article, a-posteriori error estimates for finite element solutions are derived in an asymptotic form for h 0 where h measures the size of the elements.
Abstract: Computable a-posteriori error estimates for finite element solutions are derived in an asymptotic form for h 0 where h measures the size of the elements. The approach has similarity to the residual method but differs from it in the use of norms of negative Sobolev spaces corresponding to the given bilinear (energy) form. For clarity the presentation is restricted to one-dimensional model problems. More specifically, the source, eigenvalue, and parabolic problems are considered involving a linear, self-adjoint operator of the second order. Generalizations to more general one-dimensional problems are straightforward, and the results also extend to higher space dimensions; but this involves some additional considerations. The estimates can be used for a practical a-posteriori assessment of the accuracy of a computed finite element solution, and they provide a basis for the design of adaptive finite element solvers.
1,211 citations
References
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Book•
01 Jan 1975
1,287 citations
TL;DR: In this paper, a study is made of optimum grids in two dimensional plane stress problems from which specific guidelines are suggested such that near-optimal grids can be selected by the analyst.
69 citations
TL;DR: In this article, it is shown that a true minimum of the system potential energy must consider the idealization geometry as a primary parameter, and a technique for determining the optimum solution is described and is applied to an elementary two-dimensional example.
37 citations
TL;DR: The results of P. Henriei on the asymptotic behavior of the tnmcation error are used in order to get the simpler problem, and the functions involved are sufficiently smooth so that the results of Henrici are valid.
Abstract: In the integration of a system of ordin'~ry differential equations, the simplest approach is to use a fixed step size. However, over some parts of the range of i~tegra~ion it: is generally possible to ~ake a larger step size wilhoue, seriously affecd~g the \"local ~nmcatiou error.\" This gives rise to the currently popular \"halving and doubling\" me~hod, in which one changes the step size ir~ such a way as to keep the local truncation error more or less constant. This, however, is not necessarily optimal since a smM1 local truncation error in some parts of the range of integration can give rise to a large total truncation error. The basic problem is then to choose the step size in an optimal way; i.e. for a fixed mmff)er of mesh points, how should one distribute the mesh points in o:der to achieve the smallest error ~ at the end of the range of integration. (One might ask instead that the integral of the square of the truncation error over the range of integration should be minimized instead of the error at the end of the interval. This is reasonable when the error over the entire range is of interest instead of simply the error at the end. This problem does not seem to have a simple closed fonn solution and we will not discuss it here.) The problem in this generality is extremely diffmult; hence we will first approximate it, by a simpler problem, and we will solve the simpler problem eom-pletely. Specifically, we use the results of P. Henriei [1] on the asymptotic behavior of the tnmcation error in order to get the simpler problem. In order to solve ~he simpler problem we make the following assumptions: (1 There is only one differential equation. (Otherwise, a simple closed form solution such as given here does not seem to exist; instead one has an unpleasant integral equation to solve.) (2) The (approximate) local truncation error has one sign throughout the range of integration. (Otherwise, the solution becomes very strange; one may find that it is necessary to make as large an error as possible over some parts of the range of integration.) (3) The functions involved are sufficiently smooth so that the results of Henrici are valid. We will also make some further smoothness assumptions as we go along. For practical investigation one can sometimes weaken these assumptions. Thus if there …
22 citations
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..., [13], [14])....
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01 Dec 1975
TL;DR: Present programs for finite-element analysis require the user to make numerous, critical a priori decisions that influence strongly the accuracy and reliability of the results, the cost of the computation, and other related factors.
Abstract: Present programs for finite-element analysis require the user to make numerous, critical a priori decisions. They often represent difficult mathematical problems and may influence strongly the accuracy and reliability of the results, the cost of the computation, and other related factors. This paper discusses some of these decisions and their mathematical aspects in the case of several typical examples. More specifically, the questions addressed here concern the effect of different mathematical formulations of the basic problem upon the results, the influence of the desired accuracy on the efficiency of the process, the selection and comparison of different types of elements, and for nonlinear problems the choice of efficient methods for solving the resulting finite-dimensional equations. In all cases a consistent use of self-adaptive techniques is strongly indicated. (auth)
12 citations
"Error Estimates for Adaptive Finite..." refers background in this paper
...The concepts leading to our estimates may also be applied to the estimation of the formulation error of the problem itself in comparison to a "higher" problem (see also [10])....
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