Everything you always wanted to know about a-posteriori error estimation in finite element methods, but were afraid to ask

In this paper the basic concepts to obtain a posteriori error estimates for the finite element method are reviewed. Explicit residual-based, implicit (namely subdomain and element) residual-based, hierarchical-based, recovery-base and functional-based error estimators as well as goal oriented error estimators are presented for a test elliptic boundary value problem. These notes are an introductory presentation, reviewing in a not-too-technical way the fundamental concepts involved in the subject and do not aim at being exhaustive or complete but rather simple and easy to follow. For more detailed explanations, we refer the interested reader to [3] and eventually to [4],[9],[15],[27],[30], chapter 4 of [37],[38],[39] – and the references therein – where most of the material contained in this report can be found

A numerical study of an adaptive finite element method of lines approach for coupled reaction-diffusion equations in Omega - partialOmega.

A numerical study of an adaptive finite element method of lines (AFEMOL) approach is presented for the approximation of the solution of a system of reaction-diffusion equations coupling species defined on a 2-dimensional domain Ω and species confined to the boundary of the domain ∂Ω. In order to bound the energy norm of the space discretization error, in the AFEMOL the spatial mesh changes automatically at selected times when the underlying triangulation is refined in areas where it is needed. The decision of when and where to modify the mesh is based on the estimation of the space discretization error. The adaptive process and the a-posteriori explicit error estimation exploited in this note are a modification of the pioneer work developed by Bieterman and Babuschka in [Numer. Math. 40 (1982), 339], [Numer. Math. 40 (1982), 373], [J. Comput. Phys. 63 (1986), 33]. The primary interest, in the manuscript, is the effect of the coupling Ω–∂Ω on the performance of the error estimator and the successive adaptive process. Our numerical results indicate that the global error estimators are accurate, the local error indicators are reliable and that the adaptive strategy successfully controls the space discretization error

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Abstract. We study a number of aspects of the colonisation of the Eastern Scheldt by the Japanese Oyster. We formulate and analyse some simple models of the spatial spreading, and determine a rough dependence of the spreading behaviour on parameters. We examine the suggestion of reducing salinity by opening freshwater dams, with the aim of reducing oyster fertility, and make predictions of the effect of such measures. Finally, we present an outline of a large-scale simulation taking into account detailed data on the geometry and sea floor properties of the Eastern Scheldt