By Repin, Sergey
This e-book 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 accordance 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 sensible research. those estimates are acceptable to approximate options computed by means of quite a few methods. Read more...
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Additional resources for A posteriori estimates for partial differential equations
13) is understood in the sense of distributions. Formally, this idea can be extended to a wide class of linear problems. Indeed, if A W X ! v/ D f C Av is the residual. 18)) may be a difficult task (because, in general, the error functional is a distribution so that we have a boundary value problem with rather irregular righthand side). Moreover, the accuracy of an approximate solution obviously affects the accuracy of error estimation, so that a new error estimation problem arises. In practical applications, several methods are used in order to overcome the above mentioned difficulties.
We discuss properties of the estimates, their practical implementation, and relationships between them and a posteriori estimates of other types. v/ D jrvj f v dx: v2V0 2 Henceforth, we call it Problem P . rv; y/ D y2Y 1 2 jyj f v 2 Ã dx; where Y D L2 . ; Rd /. x/. 2. P / related to each other? x; y/. 1. x; y/ be a functional defined on the elements of two nonempty sets X and Y . x; y/: y2Y x2X x2X y2Y Proof. x; y/ inf L. ; y/; 2X 8x 2 X; y 2 Y: Pass to the supremum over y 2 Y . x; y/ y2Y sup inf L.
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.
A posteriori estimates for partial differential equations by Repin, Sergey