A review of a posteriori error estimation and adaptive mesh-refinement techniques
Wiley-Teubner Series Advances in Numerical Mathematics Editors Hans Georg Bock Mitchell Luskin Wolfgang Hackbusch Rolf Rannacher A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques Rüdiger Verfürth Ruhr-Universität Bochum, Germany Self-adaptive discretization methods have gained an enormous importance for the numerical solution of partial differential equations which arise in physical and technical applications. The aim of these methods is to obtain a numerical solution within a prescribed tolerance using a minimal amount of work. The main tools utilised are a posteriori error estimators and indicators which are able to give global and local information on the error of the numerical solution, using only the computed numerical solution and known data of the problem. Presenting the most frequently used error estimators which have been developed by various scientists in the last two decades, this book demonstrates that they are all based on the same basic principles. These principles are then used to develop an abstract framework which is able to handle general nonlinear problems. The abstract results are applied to various classes of nonlinear elliptic partial differential equations from, for example, fluid and continuum mechanics, to yield reliable and easily computable error estimators. The book covers stationary problems but omits transient problems, where theory is often still too complex and not yet well developed.
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A Simple Model Problem
Abstract Nonlinear Equations
4 other sections not shown
auxiliary problems Banach space biharmonic equation Cauchy-Schwarz inequality circular segment computed Consider an arbitrary consistency error define denote depend Dirichlet Dirichlet boundary conditions discrete problem dual space eigenvalue elementwise Equation 1.9 equivalent error estimates hold estimate of Proposition Example exterior normal F(uh finite element spaces following a posteriori Fredholm operator functions given by Equations Gram determinant hanging nodes higher order imply Jn Jn Lemma linear finite elements Lipschitz continuous lower bounds marked edge mesh Navier-Stokes equations Neumann boundary conditions norm numerical solution obtain operator piecewise linear polynomial degree posteriori error estimates Problem 1.1 proof Proposition 2.1 refined regularly refinement strategy regular Remark results of Section satisfied Section 3.1 smallest angle solution of Problem space Yh superconvergence tetrahedron triangle unique solution vanishes vertex vertices Vufc Vuft weak solution yield