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21
A NEW MULTISCALE FINITE ELEMENT METHOD FOR HIGHCONTRAST ELLIPTIC INTERFACE PROBLEMS
"... Abstract. We introduce a new multiscale finite element method which is able to accurately capture solutions of elliptic interface problems with high contrast coefficients by using only coarse quasiuniform meshes, and without resolving the interfaces. A typical application would be the modelling of f ..."
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Abstract. We introduce a new multiscale finite element method which is able to accurately capture solutions of elliptic interface problems with high contrast coefficients by using only coarse quasiuniform meshes, and without resolving the interfaces. A typical application would be the modelling of flow in a porous medium containing a number of inclusions of low (or high) permeability embedded in a matrix of high (respectively low) permeability. Our method is H 1 conforming, with degrees of freedom at the nodes of a triangular mesh and requires the solution of subgrid problems for the basis functions on elements which straddle the coefficient interface, but uses standard linear approximation otherwise. A key point is the introduction of novel coefficientdependent boundary conditions for the subgrid problems. Under moderate assumptions, we prove that our methods have (optimal) convergence rate of O(h) in the energy norm and O(h 2) in the L2 norm where h is the (coarse) mesh diameter and the hidden constants in these estimates are independent of the “contrast ” (i.e. ratio of largest to smallest value) of the PDE coefficient. For standard elements the best estimate in the energy norm would be O(h 1/2−ε) with a hidden constant which in general depends on the contrast. The new interior boundary conditions depend not only on the contrast of the coefficients, but also on the angles of intersection of the interface with the element edges. 1.
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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Cited by 26 (11 self)
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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.
Analysis of a twolevel Schwarz method with coarse spaces based on local DirichlettoNeumann maps
 Comput. Methods Appl. Math
"... Schwarz method with coarse spaces based on local Dirichlet–to–Neumann maps. computer ..."
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Schwarz method with coarse spaces based on local Dirichlet–to–Neumann maps. computer
Analysis of FETI methods for multiscale PDEs  Part II: Interface variation
, 2009
"... In this article we give a new rigorous condition number estimate of the finite element tearing and interconnecting (FETI) method and a variant thereof, allfloating FETI. We consider the scalar elliptic equation in a two or threedimensional domain with a highly heterogeneous (multiscale) diffusi ..."
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In this article we give a new rigorous condition number estimate of the finite element tearing and interconnecting (FETI) method and a variant thereof, allfloating FETI. We consider the scalar elliptic equation in a two or threedimensional domain with a highly heterogeneous (multiscale) diffusion coefficient. This coefficient is allowed to have large jumps not only across but also along subdomain interfaces and in the interior of the subdomains. In other words, the subdomain partitioning does not need to resolve any jumps in the coefficient. Under suitable assumptions, we can show that the condition numbers of the onelevel and the allfloating FETI system are robust with respect to strong variations in the contrast in the coefficient. We get only a dependence on some geometric parameters associated with the coefficient variation. In particular, we can show robustness for socalled face, edge, and vertex islands in highcontrast media. As a central tool we prove and use new weighted Poincaré and discrete Sobolev type inequalities that are explicit in the weight. Our theoretical findings are confirmed in a series of numerical experiments.
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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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.
Adaptive BDDC in Three Dimensions
, 2009
"... The adaptive BDDC method is extended to the selection of face constraints in three dimensions. A new implementation of the BDDC method is presented based on a global formulation without an explicit coarse problem, with massive parallelism provided by a multifrontal solver. Constraints are implemente ..."
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The adaptive BDDC method is extended to the selection of face constraints in three dimensions. A new implementation of the BDDC method is presented based on a global formulation without an explicit coarse problem, with massive parallelism provided by a multifrontal solver. Constraints are implemented by a projection and sparsity of the projected operator is preserved by a generalized change of variables. The effectiveness of the method is illustrated on several engineering problems.
OPTIMIZED SCHWARZ AND 2LAGRANGE MULTIPLIER METHODS FOR MULTISCALE PDES
"... In this article, we formulate and analyze a twolevel preconditioner for Optimized Schwarz and 2Lagrange Multiplier methods for PDEs with highly heterogeneous (multiscale) diffusion coefficients. The preconditioner is equipped with an automatic coarse space consisting of lowfrequency modes of the ..."
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In this article, we formulate and analyze a twolevel preconditioner for Optimized Schwarz and 2Lagrange Multiplier methods for PDEs with highly heterogeneous (multiscale) diffusion coefficients. The preconditioner is equipped with an automatic coarse space consisting of lowfrequency modes of the subdomain DirichlettoNeumann maps. Under a suitable change of basis, the preconditioner is a 2 × 2 block upper triangular matrix with the identity matrix in the upperleft block. We show that the spectrum of the preconditioned system is included in the disk having center z = 1/2 and radius r = 1/2 − , where 0 < < 1/2 is a parameter that we can choose. We further show that the GMRES algorithm applied to our heterogeneous system converges in O(1/) iterations (neglecting certain polylogarithmic terms). The number can be made arbitrarily large by automatically enriching the coarse space. Our theoretical results are confirmed by numerical experiments.
1.2 Material Parameters............................... 7
, 2012
"... Dissertation im Fach Mathematik zum Erwerb des Dr. rer. nat. an der Fakultät für Mathematik ..."
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Dissertation im Fach Mathematik zum Erwerb des Dr. rer. nat. an der Fakultät für Mathematik
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