On the Advantages and Drawbacks of A Posteriori Error Estimation for Fourth-Order Elliptic Problems
In this survey contribution, we present and compare, from the viewpoint of adaptive computation, several recently published error estimation procedures for the numerical solution of biharmonic and some further fourth order elliptic problems mostly in 2D.
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On the Advantages and Drawbacks of A Posteriori Error Estimation for Fourth-Order Elliptic Problems Karel Segeth
Abstract In this survey contribution, we present and compare, from the viewpoint of adaptive computation, several recently published error estimation procedures for the numerical solution of biharmonic and some further fourth order elliptic problems mostly in 2D. In the hp-adaptive finite element method, there are two possibilities to assess the error of the computed solution a posteriori: to construct a classical analytical error estimate or to obtain a more accurate reference solution by the same procedure as the approximate solution and, from it, the computational error estimate. For the lack of space, we sometimes only refer to the notation introduced in the papers quoted. The complete hypotheses and statements of the theorems presented should also be looked for there.
8.1 Introduction Numerical computation has always been connected with some control procedures. It means that the approximate result is of primary importance, but also the error of this computed result, i.e. some norm of the difference between the exact and approximate solution brings important information. The exact solution is usually not known. This means that we can get only some estimates of the error. The development of numerical procedures has been accompanied with a priori error estimates that are very useful in theory but usually include constants that are completely unknown, in better cases can be estimated. In particular, the development of the finite element method, and its h-version and hp-version required reliable and computable estimates of the error that depend only on the approximate solution just computed, if possible. This is the means for the local mesh refinement in the h-version and, moreover, also for the increase of the polynomial degree in the pversion.
K. Segeth () Institute of Mathematics, Academy of Sciences, Prague, Czech Republic e-mail: [email protected] S. Repin et al. (eds.), Numerical Methods for Differential Equations, Optimization, and Technological Problems, Computational Methods in Applied Sciences 27, DOI 10.1007/978-94-007-5288-7_8, © Springer Science+Business Media Dordrecht 2013
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We employ a quantity called the a posteriori error indicator ηT for all triangles T of the triangulation Th and, if not defined otherwise, the error estimator ε= ηT2 , T ∈T h
see [5], in each of the estimation strategies that follow to assess the error of the approximate solution. The quality of an a posteriori error estimator is often measured by its effectivity index, i.e. the ratio of some norm of the error estimate and the true error. An error estimator is called effective if both its effectivity index and the inverse of the index remain bounded for all meshsizes of triangulations. It is called asymptotically exact if its effectivity index converges to 1 as the meshsize tends to 0. Undoubtedly, obtaining efficient and computable a posteriori error estimates is not easy. (Note that comp
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