A Posteriori Error Analysis via Duality Theory: With by Weimin Han

By Weimin Han

This quantity presents a posteriori errors research for mathematical idealizations in modeling boundary worth difficulties, specially these bobbing up in mechanical functions, and for numerical approximations of various nonlinear variational difficulties. the writer avoids giving the consequences within the so much common, summary shape in order that it really is more uncomplicated for the reader to appreciate extra truly the fundamental principles concerned. Many examples are integrated to teach the usefulness of the derived mistakes estimates.

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This quantity is acceptable for researchers and graduate scholars in utilized and computational arithmetic, and in engineering.

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Example text

Moreover, since the bilinear form a ( . 46) is equivalent to minimizing the energy functional 1 E(v) = -a(v,v) - t(v) j(v) 2 over the space V. 26. ) implies + f o r a n y u , ~E V andandt E [ O , l ] . Variational inequality formulations of many other contact problems can be found in [94, 811. 8. FINITE ELEMENT METHOD, ERROR ESTIMATES Weak formulations of boundary value problems are the basis for development of Galerkin methods, a general framework for approximation of variational problems, that include the finite element method as a special case.

A ( u ,v ) = a ( v ,u ) for u , v E V ) , which is the case for the problems considered in this work, we can consider the problem min E ( v ) vEV where E : V -+ is an energy functional. 30) with We now look at two concrete examples. 26 ( A N OBSTACLE PROBLEM) A representative example of the elliptic variational inequality of the first kind is given by the obstacle problem. The problem is to determine the equilibrium position of an elastic membrane passing through the boundary of a planar domain, lying above an obstacle of height $, and being subject to the action of a vertical force of density T f , here 7 is the elastic tension of the membrane, and f is a given function.

The influence of the paper on later researches on the topic is tremendous. , from L~ estimates to Lp estimates for any p E (1,m), from single equations to systems, etc. The interested reader is referred to [107]. More recent comprehensive references on the topic are [99, 1001. In the literature, one may find many other results on the topic. , theoretical and computational aspects of singularities for elasticity systems are studied. The papers [92] and [I231 are devoted to a study of regularity of solutions of Stokes problems in a polygon.

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