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19
A general approach to analyse preconditioners for twobytwo block matrices
"... Twobytwo block matrices arise in various applications, such as in domain decomposition methods or, more generally, when solving boundary value problems discretized by finite elements from the separation of the node set of the mesh into ’fine’ and ’coarse’ nodes. Matrices with such a structure, in ..."
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Cited by 11 (6 self)
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Twobytwo block matrices arise in various applications, such as in domain decomposition methods or, more generally, when solving boundary value problems discretized by finite elements from the separation of the node set of the mesh into ’fine’ and ’coarse’ nodes. Matrices with such a structure, in saddle point form arise also in mixed variable finite element methods and in constrained optimization problems. A general algebraic approach to construct, analyse and control the accuracy of preconditioners for matrices in twobytwo block form is presented. This includes both symmetric and nonsymmetric matrices, as well as indefinite matrices. The action of the preconditioners can involve elementbyelement approximations and/or geometric or algebraic multigrid/multilevel
Additive schur complement approximation and application to multilevel preconditioning
 Johann Radon Institute for Computational and Applied Mathematics (RICAM) of the Austrian Academy of Sciences
, 2011
"... Abstract. In the present paper we introduce an algorithm for Additive Schur Complement Approximation (ASCA). This approximation technique can be applied in various iterative methods for solving systems of linear algebraic equations arising from finite element (FE) discretization of Partial Different ..."
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Cited by 8 (3 self)
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Abstract. In the present paper we introduce an algorithm for Additive Schur Complement Approximation (ASCA). This approximation technique can be applied in various iterative methods for solving systems of linear algebraic equations arising from finite element (FE) discretization of Partial Differential Equations (PDE). Here we will show how the ASCA can be used to construct a nonlinear Algebraic MultiLevel Iteration (AMLI) method. The construction involves a linear (multiplicative) twolevel preconditioner at each level, which is computed in the course of a simultaneous exact twobytwo block factorization of local (stiffness) matrices associated with a covering of the entire domain by overlapping subdomains. Unlike in Schwarz type domain decomposition methods this method does not require a global coarse problem but instead uses local coarse problems to provide global communication. We prove a robust condition number bound and present numerical tests that demonstrate that the ASCA when combined with a proper AMLIcycle results in a multilevel method of optimal order of computational complexity. 1.
On an augmented Lagrangianbased preconditioning of Oseen type problems
"... The paper deals with a general framework for constructing preconditioners for saddle point matrices, in particular as arising in the discrete linearized NavierStokes equations (Oseen’s problem). We utilize the socalled augmented Lagrangian approach, where the original linear system of equations is ..."
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Cited by 7 (6 self)
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The paper deals with a general framework for constructing preconditioners for saddle point matrices, in particular as arising in the discrete linearized NavierStokes equations (Oseen’s problem). We utilize the socalled augmented Lagrangian approach, where the original linear system of equations is first transformed to an equivalent one, which latter is then solved by a preconditioned iterative solution method. The matrices in the linear systems, arising after the discretization of Oseen’s problem, are of twobytwo block form as are the best known preconditioners for these. In the augmented Lagrangian formulation, a scalar regularization parameter is involved, which strongly influences the quality of the blockpreconditioners for the system matrix (referred to as outer), as well as the conditioning and the solution of systems with the resulting pivot block (referred to as inner) which, in the case of large scale numerical simulations has also to be solved using an iterative method. We analyse the impact of the value of the regularization parameter on the convergence of both outer and inner solution methods. The particular preconditioner used in this work exploits the inverse of the pressure mass matrix. We study the effect of various approximations of that inverse on the performance of the preconditioners, in particular that of a sparse approximate inverse, computed in an elementbyelement fashion. We analyse and compare the spectra of the preconditioned matrices for the different approximations and show that the resulting preconditioner is independent of problem, discretization and method parameters, namely, viscosity, mesh size, mesh anisotropy. We also discuss possible approaches to solve the modified pivot matrix block. Keywords: NavierStokes equations, saddle point systems, augmented Lagrangian, finite elements, approximation of mass matrixiterative methods, preconditioning 1
Bängtsson E. Preconditioning of nonsymmetric saddle point systems as arising in modelling of viscoelastic problems
 ETNA
"... Abstract. In this paper we consider numerical simulations of the socalled glacial rebound phenomenon and the use of efficient preconditioned iterative solution methods in that context. The problem originates from modeling the response of the solid earth to large scale glacial advance and recession ..."
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Cited by 6 (1 self)
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Abstract. In this paper we consider numerical simulations of the socalled glacial rebound phenomenon and the use of efficient preconditioned iterative solution methods in that context. The problem originates from modeling the response of the solid earth to large scale glacial advance and recession which may have provoked very large earthquakes in Northern Scandinavia. The need for such numerical simulations is due to ongoing investigations on safety assessment of radioactive waste repositories. The continuous setting of the problem is to solve an integrodifferential equation in a large timespace domain. This problem is then discretized using a finite element method in space and a suitable discretization in time, and gives rise to the solution of a large number of linear systems with nonsymmetric matrices of saddle point form. We outline the properties of the corresponding linear systems of equations, discuss possible preconditioning strategies, and present some numerical experiments.
Finite element blockfactorized preconditioners
, 2007
"... In this work we consider blockfactorized preconditioners for the iterative solution of systems of linear algebraic equations arising from finite element discretizations of scalar and vector partial differential equations of elliptic type. For the construction of the preconditioners we utilize a gen ..."
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Cited by 4 (2 self)
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In this work we consider blockfactorized preconditioners for the iterative solution of systems of linear algebraic equations arising from finite element discretizations of scalar and vector partial differential equations of elliptic type. For the construction of the preconditioners we utilize a general twolevel standard finite element framework and the corresponding block twobytwo form of the system matrix, induced by a splitting of the finite element spaces, referred
ROBUST MULTILEVEL METHODS FOR QUADRATIC FINITE ELEMENT ANISOTROPIC ELLIPTIC PROBLEMS
"... Abstract. This paper discusses a class of multilevel preconditioners based on approximate block factorization for conforming finite element methods (FEM) employing quadratic trial and test functions. The main focus is on diffusion problems governed by a scalar elliptic partial differential equation ..."
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Abstract. This paper discusses a class of multilevel preconditioners based on approximate block factorization for conforming finite element methods (FEM) employing quadratic trial and test functions. The main focus is on diffusion problems governed by a scalar elliptic partial differential equation (PDE) with a strongly anisotropic coefficient tensor. The proposed method provides a high robustness with respect to nongridaligned anisotropy, which is achieved by the interaction of the following components: (i) an additive Schur complement approximation to construct the coarsegrid operator; (ii) a global block (Jacobi or GaussSeidel) smoother complementing the coarsegrid correction based on (i); and (iii) utilization of an augmented coarse grid, which enhances the efficiency of the interplay between (i) and (ii); The performed analysis indicates the high robustness of the resulting twolevel method. Moreover, numerical tests with a nonlinear algebraic multilevel iteration (AMLI) method demonstrate that the presented twolevel method can be applied successfully in the recursive construction of uniform multilevel preconditioners of optimal or nearly optimal order of computational complexity. 1.
Robust Preconditioned Iterative Solution Methods for Largescale Nonsymmetric Problems
, 2005
"... We study robust, preconditioned, iterative solution methods for largescale linear systems of equations, arising from different applications in geophysics and geotechnics. The first type of linear systems studied here, which are dense, arise from a boundary element type of discretization of crack pro ..."
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We study robust, preconditioned, iterative solution methods for largescale linear systems of equations, arising from different applications in geophysics and geotechnics. The first type of linear systems studied here, which are dense, arise from a boundary element type of discretization of crack propagation in brittle material. Numerical experiment show that simple algebraic preconditioning strategies results in iterative schemes that are highly competitive with a direct solution method. The second type of algebraic systems are nonsymmetric and indefinite and arise from finite element discretization of the partial differential equations describing the elastic part of glacial rebound processes. An equal order finite element discretization is analyzed and an optimal stabilization parameter is derived. The indefinite algebraic systems are of 2by2block form, and therefore block preconditioners of blockfactorized or blocktriangular form are
Automatic Mesh Generation for 3D Objects. February 1996
, 1997
"... A Mixed Variational Formulation for 3D Magnetostatics and its Finite Element February 1996 Discretisation. 964 Bodo Heise und Michael Jung Robust Parallel NewtonMultilevel Methods. February 1996 ..."
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A Mixed Variational Formulation for 3D Magnetostatics and its Finite Element February 1996 Discretisation. 964 Bodo Heise und Michael Jung Robust Parallel NewtonMultilevel Methods. February 1996