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Robust domain decomposition preconditioners for abstract symmetric positive definite bilinear forms. Arxiv preprint arXiv:1105.1131
, 2011
"... Abstract. An abstract framework for constructing stable decompositions of the spaces corresponding to general symmetric positive definite problems into “local ” subspaces and a global “coarse ” space is developed. Particular applications of this abstract framework include practically important probl ..."
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Abstract. An abstract framework for constructing stable decompositions of the spaces corresponding to general symmetric positive definite problems into “local ” subspaces and a global “coarse ” space is developed. Particular applications of this abstract framework include practically important problems in porous media applications such as: the scalar elliptic (pressure) equation and the stream function formulation of its mixed form, Stokes’ and Brinkman’s equations. The constant in the corresponding abstract energy estimate is shown to be robust with respect to mesh parameters as well as the contrast, which is defined as the ratio of high and low values of the conductivity (or permeability). The derived stable decomposition allows to construct additive overlapping Schwarz iterative methods with condition numbers uniformly bounded with respect to the contrast and mesh parameters. The coarse spaces are obtained by patching together the eigenfunctions corresponding to the smallest eigenvalues of certain local problems. A detailed analysis of the abstract setting is provided. The proposed decomposition builds on a method of Efendiev and Galvis [Multiscale Model. Simul., 8 (2010), pp. 1461–1483] developed for second order scalar elliptic problems with high contrast. Applications to the finite element discretizations of the second order elliptic problem in Galerkin and mixed formulation, the Stokes equations, and Brinkman’s problem are presented. A number of numerical experiments for these problems in two spatial dimensions are provided. 1.
Scaling Up through Domain Decomposition
, 2009
"... In this paper 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 homogenisation theory is not applicable. ..."
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Cited by 6 (3 self)
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In this paper 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 homogenisation 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 onelevel FETI, are proposed that are robust to strong coefficient variation. Moreover, we extend the theoretical results for the first time also to the dualprimal variant of FETI.
New theoretical coefficient robustness results for FETIDP
 Submitted Proc DD20
, 2011
"... In this short note, we present new weighted Poincaré inequalities (WPIs) with weighted averages that allow a robustness analysis of dualprimal finite element tearing and interconnecting (FETIDP) methods in certain cases where jumps of coefficients are not aligned with the subdomain partition. ..."
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Cited by 2 (1 self)
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In this short note, we present new weighted Poincaré inequalities (WPIs) with weighted averages that allow a robustness analysis of dualprimal finite element tearing and interconnecting (FETIDP) methods in certain cases where jumps of coefficients are not aligned with the subdomain partition.
Domain Decomposition and Upscaling
, 2009
"... In this talk we discuss the use of domain decomposition parallel iterative solvers for highly heterogeneous problems of flow in porous media, in both the deterministic and (MonteCarlo simulated) stochastic cases. We are particularly interested in the case of highly unstructured coefficient variatio ..."
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In this talk we discuss the use of domain decomposition parallel iterative solvers for highly heterogeneous problems of flow in porous media, in both the deterministic and (MonteCarlo simulated) stochastic cases. We are particularly interested in the case of highly unstructured coefficient variation where standard periodic or stochastic homogenisation theory is not applicable, and where there is no a priori scale separation. We will restrict attention to the important model elliptic problem (1) − ∇ · (K∇u) = f, in a bounded polygonal or polyhedral domain Ω ⊂ R d, d = 2, 3, with suitable boundary data on the boundary ∂Ω. The d × d coefficient tensor K(x) is assumed symmetric positive definite, but may vary over many orders of magnitude in an unstructured way on Ω. Many examples arise in groundwater flow and oil reservoir modelling, e.g. in the context of the SPE10 benchmark or in Monte Carlo simulations of stochastic models for strong heteoregeneities [3] (see Figure 1). Let T h
Problems
, 2012
"... A Mixed Variational Formulation for 3D Magnetostatics and its Finite Element February 1996 ..."
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A Mixed Variational Formulation for 3D Magnetostatics and its Finite Element February 1996
endorsement purposes. MULTILEVEL METHODS FOR ELLIPTIC PROBLEMS WITH HIGHLY VARYING COEFFICIENTS ON NONALIGNED COARSE GRIDS
, 2010
"... This document was prepared as an account of work sponsored by an agency of the United States ..."
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This document was prepared as an account of work sponsored by an agency of the United States
Parnum11 Gordon and Gordon AN OVERVIEW OF THE CARPCG ALGORITHM
"... Recent work shows that the blockparallel CARPCG algorithm [Parallel Computing 36, 2010] is extremely effective on sparse nonsymmetric linear systems with very small diagonal elements, including cases with discontinuous coefficients. In contrast to most known solvers, the effectiveness of CARPCG ..."
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Recent work shows that the blockparallel CARPCG algorithm [Parallel Computing 36, 2010] is extremely effective on sparse nonsymmetric linear systems with very small diagonal elements, including cases with discontinuous coefficients. In contrast to most known solvers, the effectiveness of CARPCG often improves as the diagonal elements become smaller. This property is shown to follow from the foundations on which CARPCG is based. The unique behavior of CARPCG is demonstrated with some old and new results. Previously studied problems consist of convectiondominated elliptic PDEs, with and without discontinuous coefficients. New results using highfrequency Helmholtz equations in the heterogeneous Marmousi model, discretized with 2nd, 4th and 6th order finite difference schemes, indicate the insufficiency of loworder schemes and the good parallel scalability of CARPCG at high frequencies. 1