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Constantin Bacuta. James Bramble. Regularity estimates for elliptic boundary value problems in Besov spaces. New regularity estimates for its solution in terms of Besov and Sobolev norms of fractional order are proved. The analysis is based on new inter- polation results and multilevel representations of norms on Sobolev and Besov spaces. The results can be extended to a large class of elliptic boundary value problems.

Some new sharp finite element error estimates are deduced. Introduction Regularity estimates of the solutions of elliptic boundary value problems in terms of Sobolev norms of fractional order are known as shift theorems or shift estimates. The shift estimates for the Laplace operator with Dirichlet boundary conditions on nonsmooth domains are well known see, e. Here the norms involved are the standard Sobolev fractional norms. The new shift result leads to new finite element convergence estimates.

The method presented here can be extended to other boundary conditions such as Neumann or mixed Neumann-Dirichlet conditions. We are led to the following type of interpolation problem. Key words and phrases. Our approach in proving subspace interpolation problems is to use multilevel representations of the norms for the Sobolev spaces of integer order. For multilevel representations of norms see, e. In Section 3 the main result concerning the codimension-one subspace interpolation problem is presented.

Boundary Conditions Replace Initial Conditions

Shift theorems for the Poisson equation on polygonal domains are considered in Section 4. In the last section, a straightforward application of the above interpolation results is shown to lead to some new estimates for finite element approximations. Interpolation results In this section we give some basic definitions and results concerning Besov spaces and Hilbert spaces as interpolation between Hilbert spaces using the real method of interpolation of Lions and Peetre see [3], [4] , [21]. Interpolation between Hilbert spaces. Let D S denote the subset of X consisting of all elements u such that 2.

For any u in D S the anti-linear form 2. Then by Riesz representation theorem, there exists an element Su in Y such that 2.

In this way S is a well defined operator with domain D S in Y. The next result illustrates the properties of S see [21]. Proposition 2. The next lemma provides the relation between K t, u and the connecting operator S.

2d Finite Difference Method

Lemma 2. Then 2. XU Remark 2. Interpolation between subspaces of a Hilbert space. To apply the interpolation results from the previous section we need to check that the density part of the condition 2. The proof can be found in [2].

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For completeness we include the proof also. First let us assume that the condition 2. Hence XK fails to be dense in Y. This is exactly the opposite of 2. The proof of 2. Next, 2. Theorem 2. Let u be fixed in X and set 2. From 2. XU Combining 2. With the notation adopted in 2. Then, 2. Such an extension would be needed, for example, to treat the case of mixed Neumann-Dirichlet conditions.

Proofs for the multilevel representation of the norm on H 1 , for specific choices of the spaces Mk can be found in [2], [10], [25] and [27]. Scales of multilevel norms. By using 2. Lemma 3.

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Subspace interpolation results. Then the following conditions are also satisfied. The constants involved in this proof might change at different occurrences. Using the identity 3. Finally, C. According to Lemma 2.

XU From C. Let us further note here that by proving 3. Then, using 2. Therefore, 3. It is not difficult to see that the embedding stated in Theorem 3. It is known see [2] that having C. Using Lemma 3. Thus one might expect that 3. A weaker result, but in some sense a better result, compared with Theorem 3.

In addition, Theorem 3. Theorem 3. According to Corollary 3. Applications to shift theorem for the Laplace operator on polygonal domains. In this section we will prove that the estimate 1. Shift estimate for the critical value. The variational formulation of 1. Then, see e. In fact we have that 4. Grisvard characterized the orthog- onal complement N of the range of the Laplace operator for the case of a polygonal domain see[15], [16]. Thus, by interpolation, we have 4.

Then, we have the fol- lowing theorem. Theorem 4. Let u be the variational solution of 1. XU Proof. Use 4.