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Atmospheric tomography is a prerequisite step before wavefront perturbations can be corrected in Multi Conjucate Adaptive Optics (MCAO) systems. Recently, the use of a Landweber-Kaczmarz iteration has been proposed to solve the tomography problem. However, due to the geometric partitioning of the update steps in this method, discontinuities in the solution appear that are physically implausible. We investigate augmenting the Landweber-Kaczmarz iteration with a smoothing step based on solving an elliptic partial differential equation to alleviate these discontinuities. [ABSTRACT FROM AUTHOR]

Among the well-known constants in the theory of boundary integral equations are the coercivity constants of the single-layer potential and the hypersingular boundary integral operator, and the contraction constant of the double-layer potential. Whereas there have been rigorous studies how these constants depend on the size and aspect ratio of the underlying domain, only little is known on their dependency on the shape of the boundary. In this article, we consider the homogeneous Laplace equation and derive explicit estimates for the above-mentioned constants in three dimensions. Using an alternative trace norm, we make the dependency explicit in two geometric parameters, the so-called Jones parameter and the constant in Poincaré's inequality. The latter one can be tracked back to the constant in an isoperimetric inequality. There are many domains with quite irregular boundaries, where these parameters stay bounded. Our results provide a new tool in the analysis of numerical methods for boundary integral equations and in particular for boundary element based domain decomposition methods. [ABSTRACT FROM PUBLISHER]

Poincaré-type inequalities are a key tool in the analysis of partial differential equations. They play a particularly central role in the analysis of domain decomposition and multilevel iterative methods for second-order elliptic problems. When the diffusion coefficient varies within a subdomain or within a coarse grid element, then condition number bounds for these methods based on standard Poincaré inequalities may be overly pessimistic. In this paper, we present new results on weighted Poincaré-type inequalities for very general classes of coefficients that lead to sharper bounds independent of any possible large variation in the coefficients. The main requirement on the coefficients is some form of quasi-monotonicity that we will carefully describe and analyse. The Poincaré constants depend on the topology and the geometry of regions of relatively high and/or low coefficient values, and we shall study these dependencies in detail. Applications of the inequalities in the analysis of domain decomposition and multigrid methods can be found in Pechstein & Scheichl (2011, Numer. Math., 118) and Scheichl et al. (2012, SIAM J. Numer. Anal., 50). [ABSTRACT FROM PUBLISHER]

Abstract: Finite element tearing and interconnecting (FETI) methods and boundary element tearing and interconnecting (BETI) methods are special iterative substructuring methods with Lagrange multipliers. For elliptic boundary value problems on bounded domains, the condition number of these methods can be rigorously bounded by , where H is the subdomain diameter and h the mesh size. The constant C is independent of H, h and possible jumps in the coefficients of the partial differential equation. In certain situations, e.g., in electromagnetic field computations, instead of imposing artificial boundary conditions one may be interested in modelling the real physical behaviour in an exterior domain with a radiation condition. In this work we analyze one-level BETI methods for such unbounded domains and show explicit condition number estimates similar to the one above. Our theoretical results are confirmed in numerical experiments. [Copyright &y& Elsevier]

In this article, we discuss domain decomposition parallel iterative solvers for highly heterogeneous problems of flow and transport in porous media. We are particularly interested in highly unstructured coefficient variation where standard periodic or stochastic homogenization theory is not applicable. When the smallest scale at which the coefficient varies is very small, it is often necessary to scale up the equation to a coarser grid to make the problem computationally feasible. Standard upscaling or multiscale techniques require the solution of local problems in each coarse element, leading to a computational complexity that is at least linear in the global number N of unknowns on the subgrid. Moreover, except for the periodic and the isotropic random case, a theoretical analysis of the accuracy of the upscaled solution is not yet available. Multilevel iterative methods for the original problem on the subgrid, such as multigrid or domain decomposition, lead to similar computational complexity (i.e. O(N)) and are therefore a viable alternative. However, previously no theory was available guaranteeing the robustness of these methods to large coefficient variation. We review a sequence of recent papers where simple variants of domain decomposition methods, such as overlapping Schwarz and one-level FETI, are proposed that are robust to strong coefficient variation. Moreover, we also extend the theoretical results, for the first time, to the dual-primal variant of FETI. [ABSTRACT FROM AUTHOR]

This paper analyses two-level Schwarz methods for matrices arising from the p-version finite-element method on triangular and tetrahedral meshes. The coarse level consists of the lowest-order finite-element space. On the fine level, we investigate several decompositions with large or small overlap leading to optimal or close to optimal condition numbers. The analysis is confirmed by numerical experiments for a simple model problem and an elasticity problem on a complex geometry. [ABSTRACT FROM PUBLISHER]

The present paper deals with the numerical solution of boundary value problems for nonlinear potential equations of the form –∇· [ν(|∇ u|) ∇ u] = f arising e. g. in 2D magnetic field computations. We apply Newton's method to the nonlinear variational formulation and solve the linearized problems with coupled Finite and Boundary Element Tearing and Interconnecting (FETI/BETI) methods. The spectrum of the subdomain Jacobi matrices may show high variation due to the nonlinear behavior of the reluctivity on ferromagnetic subdomains. A special preconditioner is proposed to overcome these problems. In order to get a good initial guess for the Newton iteration, we use the nested iteration strategy based on a hierarchy of nested grids. Finally, we discuss our first numerical results obtained from the numerical solution of some 2D magnetostatic model problems. [ABSTRACT FROM AUTHOR]