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The expression C:=ltiui with + 48 A POSTERIORI ERROR ANALYSIS VIA DUALITY THEORY nonnegative numbers t l , . . , tn satisfying Cy=2=1 ti = 1 is called a convex combination of the elements u l , . . , u,. DEFINITION 2 . , f ( u )and f ( v )are not simultaneously infinite with opposite signs. 3 Let K be a convex set in V avtd f : K f ( v )= +R. If { y; w v E K, v is convex, then we say f is convex on K. 1) holds for any u, v E K , u # v and t E ( 0 , l ) . 3. In other words, we will use the same symbol f for both the function defined on K and its extension by oo to the complement of K in the space V.

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 , .

63]. Theory of the finite element method for solving parabolic problems can be found in [I471 and more recently in [148]. Finally, we list a few representative engineering books on the finite element method, [20, 88, 163, 1641. The reader is referred to two historical notes [115, 1621 on the development of the finite element method. In this section, we will review some results of the finite element method. There are some basic aspects in the construction of finite element approximations. First we need a partition (or triangulation) of the domain of the differential equation into sub-domains called elements.

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