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16
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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Cited by 12 (7 self)
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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.
Analysis of a nonstandard finite element method based on boundary integral operators
 Electron. Trans. Numer. Anal
, 2010
"... Abstract. We present and analyze a nonstandard finite element method based on elementlocal boundary integral operators that permits polyhedral element shapes as well as meshes with hanging nodes. The method employs elementwise PDEharmonic trial functions and can thus be interpreted as a local Tre ..."
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Cited by 7 (3 self)
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Abstract. We present and analyze a nonstandard finite element method based on elementlocal boundary integral operators that permits polyhedral element shapes as well as meshes with hanging nodes. The method employs elementwise PDEharmonic trial functions and can thus be interpreted as a local Trefftz method. The construction principle requires the explicit knowledge of the fundamental solution of the partial differential operator, but only locally, i.e., in every polyhedral element. This allows us to solve PDEs with elementwise constant coefficients. In this paper we consider the diffusion equation as a model problem, but the method can be generalized to convectiondiffusionreaction problems and to systems of PDEs such as the linear elasticity system and the timeharmonic Maxwell equations with elementwise constant coefficients. We provide a rigorous error analysis of the method under quite general assumptions on the geometric properties of the elements. Numerical results confirm our theoretical estimates.
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.
AllFloating Coupled DataSparse Boundary and InterfaceConcentrated Finite Element Tearing and Interconnecting Methods
"... Efficient and robust tearing and interconnecting solvers for large scale systems of coupled boundary and finite element domain decomposition equations are the main topic of this paper. In order to reduce the complexity of the finite element part from O((H/h) d) to O((H/h) d−1), we use an interfacec ..."
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Cited by 2 (2 self)
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Efficient and robust tearing and interconnecting solvers for large scale systems of coupled boundary and finite element domain decomposition equations are the main topic of this paper. In order to reduce the complexity of the finite element part from O((H/h) d) to O((H/h) d−1), we use an interfaceconcentrated hp finite element approximation. The complexity of the boundary element part is reduced by datasparse approximations of the boundary element matrices. Finally, we arrive at a parallel solver whose complexity behaves like O((H/h) d−1) up to some polylogarithmic factor, where H, h, and d denote the usual scaling parameters of the subdomains, the minimal discretization parameter of the subdomain boundaries, and the spatial dimension, respectively.
Robust FETI solvers for multiscale elliptic PDEs
 In Proceedings of the 7th International Conference on Scientific Computing in Electrical Engineering (SCEE
, 2008
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IETI  Isogeometric Tearing and Interconnecting
, 2012
"... Finite Element Tearing and Interconnecting (FETI) methods are a powerful approach to designing solvers for largescaleproblems in computational mechanics. The numerical simulation problem is subdivided into a number of independent subproblems, which are then coupled in appropriate ways. NURBS (Non ..."
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Cited by 1 (0 self)
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Finite Element Tearing and Interconnecting (FETI) methods are a powerful approach to designing solvers for largescaleproblems in computational mechanics. The numerical simulation problem is subdivided into a number of independent subproblems, which are then coupled in appropriate ways. NURBS (NonUniform RationalBspline) basedisogeometricanalysis(IGA) applied tocomplex geometriesrequirestorepresentthe computational domain as a collection of several NURBS geometries. Since there is a natural decomposition of the computational domain into several subdomains, NURBSbased IGA is particularly well suited for using FETI methods. This paper proposes the new IsogEometric Tearing and Interconnecting (IETI) method, which combines the advanced solver design of FETI with the exact geometry representation of IGA. We describe the IETI framework for two classes of simple model problems (Poisson and linearized elasticity) and discuss the coupling of the subdomains along interfaces (both for matching interfaces and for interfaces with Tjoints, i.e. hanging nodes). Special attention is paid to the construction of a suitable preconditioner for the iterative linear solver used for the interface problem. We report several computational experiments to demonstrate the performance of the proposed IETI method.
DOMAIN DECOMPOSITION SOLVERS FOR THE FLUIDSTRUCTURE INTERACTION PROBLEMS WITH ANISOTROPIC ELASTICITY MODELS
"... Abstract. In this work, a twolayer coupled fluidstructurestructure interaction model is considered, which incorporates an anisotropic structure model into the fluidstructure interaction problems. We propose two domain decomposition solvers for such a class of coupled problems: a RobinRobin prec ..."
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Abstract. In this work, a twolayer coupled fluidstructurestructure interaction model is considered, which incorporates an anisotropic structure model into the fluidstructure interaction problems. We propose two domain decomposition solvers for such a class of coupled problems: a RobinRobin preconditioned GMRES solver combined with an inner DirichletNeumann iterative solver, and a RobinRobin preconditioned GMRES solver combined with an inner monolithic algebraic multigrid solver capable of handling an anisotropic compressible and nearly incompressible subproblem. 1.
NUMERICAL SIMULATION OF FLUIDSTRUCTURE INTERACTION PROBLEMS WITH HYPERELASTIC MODELS I: A PARTITIONED APPROACH
"... Abstract. In this work, we consider fluidstructure interaction simulation with nonlinear hyperelastic models in the solid part. We use a partitioned approach to deal with the coupled nonlinear fluidstructure interaction problems. We focus on handling the nonlinearity of the fluid and structure sub ..."
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Abstract. In this work, we consider fluidstructure interaction simulation with nonlinear hyperelastic models in the solid part. We use a partitioned approach to deal with the coupled nonlinear fluidstructure interaction problems. We focus on handling the nonlinearity of the fluid and structure subproblems, the nearincompressibility of materials, the stabilization of employed finite element discretization, and the robustness and efficiency of Krylov subspace and algebraic multigrid methods for the linearized algebraic equations. 1.
Recent advances and emerging applications . . .
, 2011
"... was prepared after this workshop by the organizers and participants based on the presentations and discussions at the workshop. The paper aims to review the major research achievements in the last decade, the current status, and the future directions of the BEM in the next decade. The review starts ..."
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was prepared after this workshop by the organizers and participants based on the presentations and discussions at the workshop. The paper aims to review the major research achievements in the last decade, the current status, and the future directions of the BEM in the next decade. The review starts with a brief introduction to the BEM. Then, new developments in Green’s functions, symmetric Galerkin formulations, boundary meshfree methods, and variationally based BEM formulations are reviewed. Next, fast solution methods for efficiently solving the BEM systems of equations, namely, the