By Repin, Sergey
This ebook offers with the trustworthy verification of the accuracy of approximate strategies that's one of many important difficulties in glossy utilized research. After giving an outline of the equipment built for versions in keeping with partial differential equations, the writer derives computable a posteriori errors estimates through the use of tools of the idea of partial differential equations and sensible research. those estimates are appropriate to approximate options computed by means of numerous methods. Read more...
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Extra info for A posteriori estimates for partial differential equations
Scherzer ). A further study of this strategy is presented in Jin Qi-nian and Hou Zong-yi . A posteriori estimates for linear ill-posed problems were investigated in A. Leonov . , in K. Braune , Jin Qi-nian , P. Meyer , F. Potra , K. Tsuruta and K. Ohmori , T. Yamamoto . Boundary value problems are often reformulated in terms of integral equations. , boundary element methods) are usually derived with the help of the techniques that exploits special properties of the iteration operator.
These observations suggest an idea to post-process ruh and find a close vector-valued function that satisfies some of the above-menioned properties. Formally, the principal scheme is as follows. ƒvh / is much closer to ƒu than ƒvh . ƒvh / cator of element-wise errors. ƒvh generates an efficient indi- Regularization. , see the paper by J. H. Bramble and A. H. Schatz , which is one of the earliest publications in this area). If the error caused by violations of a priori regularity properties dominates and a postprocessing operator efficiently performs regularization of approximate solutions, then one may hope that the difference between the approximate solution and its regularized (smoothed) counterpart represents the major part of the error.
Recently, new forms of the above-discussed error indicators has been derived and tested (see P. Neittaanm¨aki, S. Repin, and P. Turchin ). They do not exploit superconvergence of the adjoint solution and, therefore, can be used if adjoint meshes are not very regular. 4). 2) For this problem, we derive two-sided a posteriori estimates with the help of two different methods. The first method uses variational arguments, the second one is based on transformations of the corresponding integral identity.