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Weimin Han's A Posteriori Error Analysis via Duality Theory PDF

By Weimin Han

ISBN-10: 144193636X

ISBN-13: 9781441936363

This quantity offers a posteriori mistakes research for mathematical idealizations in modeling boundary price difficulties, in particular these coming up in mechanical purposes, and for numerical approximations of diverse nonlinear variational difficulties. the writer avoids giving the implications within the such a lot normal, summary shape in order that it's more uncomplicated for the reader to appreciate extra truly the fundamental rules concerned. Many examples are integrated to teach the usefulness of the derived errors estimates.

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Then we have the following result. T H E O R E1M . 2 3 Let f E W"P(R) and g j E ~ ~ + ~ - l l p > p (j I E ' ~N ) , be given, 1 < p < co. Denote q = p / ( p - 1 ) the conjugate exponent of p. Then, there exist numbers k j , , such that + 29 Preliminaries can be replaced by the In the theorem, the condition gj E ~~+~-llpJ'(r~) (rj). 7. AN INTRODUCTION OF ELLIPTIC VARIATIONAL INEQUALITIES The theory of variational inequalities plays an important role in the study of both the qualitative and numerical analysis of some nonlinear boundary value problems arising in mechanics and other applications.

17. For any v E Hhl ( R )the real function is defined in a neighborhood of t = 0 and has its maximum at t = 0. We observe that if we keep the value of 11 u 11 L~ fixed, then the value 11 Qu I I L 2 ( n ) A POSTERIORI ERROR ANALYSIS VIA DUALITY THEORY will be minimal when the function u does not change rapidly. So we expect an optimal function u should keep its sign. 16). So the best constant is expected to be Let us show that these formulas indeed provide the best constant. , [61]). 18 I f v E H 1 ( f l ) ,then Ivl E ~ ' ( f l and ), i Vv Vlvl = 0 -Vv i f v > 0, i f v = 0, i f v < 0.

Variational inequalities are problems involving either differential inequalities or inequality boundary conditions. Mechanics is a rich source of variational inequalities (cf. g. [125]), and some examples of problems that give rise to variational inequalities are obstacle and contact problems, plasticity and visco-plasticity problems, Stefan problems, unilateral problems of plates and shells, and non-Newtonian flows involving Bingham fluids. An early comprehensive reference on the topic is 1471, where many nonlinear boundary value problems in mechanics and physics are formulated and studied in the framework of variational inequalities.

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A Posteriori Error Analysis via Duality Theory by Weimin Han

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