By Repin, Sergey
This ebook offers with the trustworthy verification of the accuracy of approximate suggestions that is one of many relevant difficulties in smooth utilized research. After giving an outline of the tools built for types in line with partial differential equations, the writer derives computable a posteriori mistakes estimates through the use of tools of the idea of partial differential equations and useful research. those estimates are acceptable to approximate ideas computed through numerous methods. Read more...
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Extra info for A posteriori estimates for partial differential equations
However, in many cases a priori estimates of the exact solution guarantee that ƒu 2 U; where U is a subset of U . In particular, if f 2 L2 . 2) is a vector-valued function in U D H. ; div /. Moreover, very often we know that ru 2 H 1 . ; Rd / (globally or locally). Another option is to set U D Qf . 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 .
Very often gN h is a better image of Gƒu than the functions obtained by local procedures. , see C. Carstensen and S. A. Funken  and C. Carstensen and S. Bartels  where it was shown that each averaging procedure leads to a certain a posteriori estimate). -O. -C. Tai and J. Wang . In J. Wang , it was suggested the so-called “least squares surface fitting” procedure that for problems with sufficiently smooth solutions lead to a recovered function with superconvergent properties. u uh /; where u is the exact solution of a linear elliptic problem, uh is the Galerkin approximation computed on a mesh Th and Q is the L2 -projection operator on the finitedimensional space constructed on a mesh T with the help of piecewise polynomial functions of the order r 0.
In general, finding such a constant requires solving an infinite-dimensional problem on each patch. For example, 1k D diam. 3). 11) by a certain set of polynomial functions. This provides a possibility of computing an approximate value of 1k but, in such a case, the reliability of the upper bound of the error may be lost. Another unpleasant feature of the method is that 1k and 2k depend on Th and, consequently, all of them must be recalculated if one sampling is replaced by another one. 9) is derived by formal mathematical transformations that clearly overestimate j Fuh j .