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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.

Furthermore, 44 A POSTERIORI ERROR ANALYSIS VIA DUALITY THEORY assume { P h ) is a regular family ofjnite element partitions. Then there is a constant c such that for m 5 k 1, + Since the global finite element interpolant is defined piecewise in terms of local finite element interpolants, we have the next result for global finite element interpolation error estimates. 31 hold. 17). 9) (page 16). Assume R c Kt2 is a polygonal domain, f E L2( R ) . 57). Under the solution regularity assumption u E H ~ ( R(valid ) if R is convex, cf.

We then define a function space over a general element K that is the image of the reference element K under an invertible affine mapping The mapping FK is a bijection between K and K. Over the element K ,we define a finite dimensional function space X K by Since FK is an invertible affine mapping, if x is a polynomial space of certain degree, then X K is a polynomial space of the same degree. ir = v o FK. We see that v = 6 o Fil. Thus we have the relation ~ ( x=) 6 ( k ) V x E K ,2 E K ,w i t h x = FK(ii) Using the nodal points h i , 1 5 i 5 No, of K ,we can define the nodal points on K : aiK = F K ( h i ) , i = I , .

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