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June 1, 2003
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We consider the model Dirichlet problem for Poisson's equation on a plane polygonal convex domain Ω with data ƒ in a space smoother than L 2 . The regularity and the critical case of the problem depend on the measure of the maximum angle of the domain. Interpolation theory and multilevel theory are used to obtain estimates for the critical case. As a consequence, sharp error estimates for the corresponding discrete problem are proved. Some classical shift estimates are also proved using the powerful tools of interpolation theory and mutilevel approximation theory. The results can be extended to a large class of elliptic boundary value problems.

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June 1, 2003
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In this paper a duality-based a posteriori error analysis is developed for the conforming hp Galerkin finite element approximation of second-order elliptic problems. Duality arguments combined with Galerkin orthogonality yield representations of the error in arbitrary quantities of interest. From these error estimates, criteria are derived for the simultaneous adaptation of the mesh size h and the polynomial degree p . The effectivity of this procedure is confirmed by numerical tests.

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June 1, 2003
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Many boundary integral equations for exterior Dirichlet and Neumann boundary value problems for the Helmholtz equation suffer from a motorious instability for wave numbers related to interior resonances. The so-called combined field integral equations are not affected. This article presents combined field integral equations on two-dimensional closed surfaces that possess coercivity in canonical trace spaces. For the exterior Dirichlet problem the main idea is to use suitable regularizing operators in the framework of an indirect method. This permits us to apply the classical convergence theory of conforming Galerkin methods.

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A class of data-sparse hierarchical matrices (ℋ-matrices) allows to approximate nonlocal (integral) operators with almost linear complexity [22 – 31]. In the present paper, a method is described for an explicitℋ-matrix approximation to the inverse of an elliptic differential operator with piecewise constant coefficients in ℝ d . Our approach is based on the additive splitting to the corresponding Green function, which leads to the sum of an ℋ-matrix and certain correction term including the product of data-sparse matrices of different hierarchical formats. In the case of jumping coefficients with respect to conformal domain decomposition, the approximate inverse operator is obtained as a direct sum of local inverses over subdomains and the Schur complement inverse on the interface. As a by-product, we obtain an explicit approximate inverse preconditioner with the data sparsity inherited from the ℋ-matrix format.

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June 1, 2003
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We study the discretization accuracy for first-order system least squares (FOSLS) applied to Poisson's equation as a model problem. The FOSLS formulation is based on an H 1 elliptic bilinear form ℱ. Since the order of convergence of the discretization in the L 2 and H 1 norms depends on the regularity of ℱ, we examine this property in detail. We then use these results together with an Aubin-Nitsche bound to develop improved discretization error estimates.