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January 1, 2012
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This paper investigates the discretization of mixed variational formulation as, e.g., the Stokes problem by means of the hp -version of the finite element method. The system of linear algebraic equations is solved by the preconditioned Bramble-Pasciak conjugate gradient method. The development of an efficient preconditioner requires three ingredients, a preconditioner related to the components of the velocity modes, a preconditioner for the Schur complement related to the components of the pressure modes and a discrezation by a stable finite element pair which satisfies the discrete inf-sup-condition. The last condition is also important in order to obtain a stable discretization scheme. The preconditioner for the velocity modes is adapted from fast $hp$-FEM preconditioners for the potential equation. Moreover, we will prove that the preconditioner for the Schur complement can be chosen as a diagonal matrix if the pressure is discretized by discontinuous finite elements. We will prove that the system of linear algebraic equations can be solved in almost optimal complexity. This yields quasioptimal hp -FEM solvers for the Stokes problems and the linear elasticity problems. The latter are robust with respect to the contraction ratio ν. The efficiency of the presented solver is shown in several numerical examples.

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Coarse grid correction is a key ingredient in order to have scalable domain decomposition methods. For smooth problems, the theory and practice of such two-level methods is well established, but this is not the case for problems with complicated variation and high contrasts in the coefficients. In a previous study, two of the authors introduced a coarse space adapted to highly heterogeneous coefficients using the low frequency modes of the subdomain DtN maps. In this work, we present a rigorous analysis of a two-level overlapping additive Schwarz method with this coarse space, which provides an automatic criterion for the number of modes that need to be added per subdomain to obtain a convergence rate of the order of the constant coefficient case. Our method is suitable for parallel implementation and its efficiency is demonstrated by numerical examples on some challenging problems with high heterogeneities for automatic partitionings.

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January 1, 2012
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In this paper we discuss robust two-level domain decomposition preconditioners for highly anisotropic heterogeneous multiscale problems. We present a construction of several coarse spaces that employ standard finite element and multiscale basis functions and discuss techniques to reduce the dimensions of coarse spaces without sacrificing the robustness. We experimentally study the performance of the preconditioner on a variety two-dimensional test problems with channels of high anisotropy. The numerical tests confirm the robustness of the perconditioner with respect to the underlying physical parameters.

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We study regular decompositions for H (div) spaces. In particular, we show that such regular decompositions are closely related to a previously studied ``inf-sup'' condition for parameter-dependent Stokes problems, for which we provide an alternative, more direct, proof.

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January 1, 2012
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We derive defect correction scheme for constructing the sequence of polynomials of best approximation in the uniform norm to 1/x on a finite interval with positive endpoints. As an application, we consider two-level methods for scalar elliptic partial differential equation (PDE), where the relaxation on the fine grid uses the aforementioned polynomial of best approximation. Based on a new smoothing property of this polynomial smoother that we prove, combined with a proper choice of the coarse space, we obtain as a corollary, that the convergence rate of the resulting two-level method is uniform with respect to the mesh parameters, coarsening ratio and PDE coefficient variation.

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In this paper we construct efficient domain decomposition methods for solving scalar second-order elliptic boundary problems in bounded two-dimensional domains with small holes, and present some results of numerical experiments, confirming the efficiency and robustness of the proposed domain decomposition methods.

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January 1, 2012
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We present simple formulas for the definition of the basis functions on a reduced Hsieh-Clough-Tocher (rHCT) element. These formulas use the P_3 -biorthogonal basis in the master triangle and form the resulting basis with the help of the edge vectors of the triangle only. This allows for a simple and efficient algorithm to compute the stiffness matrices.

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January 1, 2012
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This paper discusses the constructive and computational presentations of several non-local norms of discrete trace functions of H¹ (Ω) and H² (Ω) defined on the boundary or interface of an unstructured grid. We transform the nonlocal norms of trace functions to local norms of certain functions defined on the whole domain by constructing isomorphic extension operators. A unified approach is used to explore several typical examples. Additionally, we also discuss exactly invertible Poincaré–Steklov operators and their discretization.