Bacuta, C., Bramble, J.H. and Pasciak, J.E., Using finite element tools in proving shift theorems for elliptic boundary value problems, Numer. Linear Algebra Appl., 10, (2003), 33--64.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....show that the L 2 norm estimates of Theorems 4.1 and 5.1 are sharp, following [26] we consider two counterexamples. A brief description of this paper is the following. In Section 2 we give, in short, known sharp regularity estimates for the exact solution of problem (1.1) and (2. 1) based on [23, 4]. In Section 3 we present the finite volume element method. In Sections 4 and 5, we analyze the finite volume element method (1.4) and derive error estimates in H L 2 and L# norm. The approach follows the one developed in [13] and uses known sharp regularity results for the solutions of ....

....u of (1.1) is played by the constant p # 2 # # . According to [23, p. 233] p , 2.3) p = p, p p # , #, any # p # , p p # , Using also the imbedding W 2 p, cf. e.g. 23, page 34] we obtain: p . Also, for problem (2. 1) we have, cf. e.g. [4]: 2.6) s = #, s 0 # 1, #, any # # #, 0 s 0 = p 0 1 = 1 # . For the more general problem (1.1) similar results hold. Let S be a vertex of ## and denote the corresponding interior angle of# by #(S) Let , be matrices such = a ij (S) i,j=1 and T ....

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Bacuta, C., Bramble, J.H. and Pasciak, J.E., Using finite element tools in proving shift theorems for elliptic boundary value problems, Numer. Linear Algebra Appl., 10, (2003), 33--64.

....regularity of the interface #. However, without Assumption III.2, we can show (3. 11) for all # [0, 1 2) Indeed we can take # h = P h #, where P h is the energy projection onto S h under the innerproduct (#, #) Since the interpolation space between H 0 and H 227 is H 1 #(## [6], we have 1 # , and thus (3.11) follows from the equivalence of i and 1 #,# . This is the only place we use Assumption III.2. Lemma III.5. Suppose that u H(curl; ## and that # h is the interpolation operator onto U h . Then, we have the following estimates. i ) for some ....

C. Bacuta, J. H. Bramble, and J. E. Pasciak, Using finite element tools in proving shift theorems for elliptic boundary value problems, Journal of Computational Linear Algebra, (2001), submitted.

....for a norm ## h defined on V V h and a constant c one can prove that #u u h # h # c#u# H 1(## , for all u # H 1 D (## , 1.1) and #u u h # h # ch#u# H 2(## , for all u # H 2 # H 1 D . 1.2) By interpolation, from (1.1) and (1. 2) we obtain that for a fixed s # [0, 1]: #u u h # h # ch s #u# [H 2(## #H 1 D(## ,H 1 D(## 1 s for all u # [H 2 # H 1 D , H 1 D(##4 1 s . If we assume that the variational solution u belongs to an intermediate space H 1 s (## # H 1 D(##1 s # (0, 1) and u is not in H 2(##3 then it is natural to ask: ....

....By definition we take [X, Y ] 0 : X and [X, Y ] 1 : Y. 3 The next lemma provides the relation between K(t, u) and the connecting operator S. Lemma 2.1. For all u # Y and t 0 , K(t, u) 2 = t 2 # (I t 2 S 1 ) 1 u, u # Y . For the proof of this lemma see, for example, [1]. Remark 2.1. Lemma 2.1 gives an alternative expression for the norm on [X, Y ] s , namely: #u# 2 [X,Y ] s : c 2 s # 0 # t 2s 1 # (I t 2 S 1 ) 1 u, u # Y dt. 2.5) In addition, by this expression for the norm (see Definition 2.1 and Theorem 15.1 in [11] it follows that ....

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C. Bacuta, J.H. Bramble and J. Pasciak. Using Finite Element Tools in Proving Shift Theorems for Elliptic Boundary Value Problems. In preparation.

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