By Repin, Sergey
This publication bargains with the trustworthy verification of the accuracy of approximate recommendations that is one of many vital difficulties in smooth utilized research. After giving an outline of the tools built for versions in line with partial differential equations, the writer derives computable a posteriori blunders estimates through the use of equipment of the idea of partial differential equations and useful research. those estimates are appropriate to approximate recommendations computed through a variety of methods. Read more...
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Additional info for A posteriori estimates for partial differential equations
Galerkin  and W. Ritz  methods using locally supported trial functions. In 60–70s, finite element methods have formed one of the main approaches to the numerical analysis of PDE’s. The foundations of the finite element method are exposed in the books by Ph. Ciarlet , S. Brenner and R. L. Scott , C. Johnson , J. T. Oden and J. N. Reddy , G. Strang and G. Fix. , O. C. Zienkiewicz and K. Morgan , and many other publications. , in the papers by I. Babuˇska  and O.
Hansbo and C. Johnson , and R. Verf¨urth . There, the reader will find detailed expositions of various approaches, results of numerical experiments, and a wide list of references. Below, 32 Chapter 2 Overview we shortly discuss several a posteriori error estimation methods developed for finite element approximations. Certainly, the exposition is not complete. Its goal is to give a view of the mathematical ideas underlying the methods. 1 Explicit residual methods From the mathematical viewpoint, the classical “residual method” is a method for finding an upper bound of the residual functional evaluated in the topology of the image space of the respective operator.
See E. Zeidler ). The major difficulty in the application of the above a posteriori estimates is that in practice it may be difficult to find a sharp upper bound of q and to establish that it is indeed less than 1. A posteriori methods for various iteration schemes have been investigated by many authors. Below we give some references that provide an idea of the results obtained and will help to find more pertinent information. In G. Auchmuty , two sided p-norm error bounds for solutions of linear systems are presented.