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Item type:Publication, Access status: without files , Preconditioned nullspace method for the Two-Dimensional Oseen problem(2009-06-12)The Oseen problem, which arises in the simulation of the time-dependent Navier- Stokes equations for incompressible fluid flow, leads to indefinite, nonsymmetric and possibly ill-conditioned linear systems of equations. This paper presents a method for obtaining a reduced linear system from the original system, which is then solved by the preconditioned BiCGStab method. The system reduction is obtained through an efficient implicit representation of a basis of discretely divergence-free functions, also known as the nullspace method. We will show a close relationship between this preconditioned nullspace method and the BFBt-preconditioned pressure Schur complement problem arising in typical block preconditioners. We will use this relationship to analyze the spectrum of the preconditioned reduced system. We will also present numerical tests for the two-dimensional Oseen problem to illustrate the performance of this method, which is similar to the performance of the BFBt-preconditioned pressure Schur complement solver. In particular, it is robust w.r.t. the strength of the viscosity for simple constant wind but not for more complicated flows, and it shows only a slight dependence on the mesh size. © 2009 Society for Industrial and Applied Mathematics.Publicationtype: Journal Article32 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: without files , Domain decomposition based H-LU preconditioning(2009-03-03); ; Hierarchical matrices provide a data-sparse way to approximate fully populated matrices. The two basic steps in the construction of an H-matrix are (a) the hierarchical construction of a matrix block partition, and (b) the blockwise approximation of matrix data by low rank matrices. In this paper, we develop a new approach to construct the necessary partition based on domain decomposition. Compared to standard geometric bisection based H-matrices, this new approach yields H-LU factorizations of finite element stiffness matrices with significantly improved storage and computational complexity requirements. These rigorously proven and numerically verified improvements result from an H-matrix block structure which is naturally suited for parallelization and in which large subblocks of the stiffness matrix remain zero in an LU factorization. We provide numerical results in which a domain decomposition based H-LU factorization is used as a preconditioner in the iterative solution of the discrete (three-dimensional) convection-diffusion equation.Publicationtype: Journal Article40 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: without files , ℋ-matrix preconditioners for symmetric saddle-point systems from meshfree discretizationMeshfree methods are suitable for solving problems on irregular domains, avoiding the use of a mesh. To deal with the boundary conditions, we can use Lagrange multipliers and obtain a sparse, symmetric and indefinite system of saddle-point type. Many methods have been developed to solve the indefinite system. Previously, we presented an algebraic method to construct an LU-based preconditioner for the saddle-point system obtained by meshfree methods, which combines the multilevel clustering method with the ℋ-matrix arithmetic. The corresponding preconditioner has both ℋ-matrix and sparse matrix subblocks. In this paper we refine the above method and propose a way to construct a pure ℋ-matrix preconditioner. We compare the new method with the old method, JOR and smoothed algebraic multigrid methods. The numerical results show that the new preconditioner outperforms the preconditioners based on the other methods.Publicationtype: Journal Article36 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: without files , Parallel black box ℋ-LU preconditioning for elliptic boundary value problems(2008-04-01); ; Hierarchical ( ℋ-) matrices provide a data-sparse way to approximate fully populated matrices. The two basic steps in the construction of an ℋ-matrix are (a) the hierarchical construction of a matrix block partition, and (b) the blockwise approximation of matrix data by low rank matrices. In the context of finite element discretisations of elliptic boundary value problems, ℋ-matrices can be used for the construction of preconditioners such as approximate ℋ-LU factors. In this paper, we develop a new black box approach to construct the necessary partition. This new approach is based on the matrix graph of the sparse stiffness matrix and no longer requires geometric data associated with the indices like the standard clustering algorithms. The black box clustering and a subsequent ℋ-LU factorisation have been implemented in parallel, and we provide numerical results in which the resulting black box ℋ-LU factorisation is used as a preconditioner in the iterative solution of the discrete (three-dimensional) convection-diffusion equation.Publicationtype: Journal Article35 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: without files , Block computation and representation of a sparse nullspace basis of a rectangular matrixIn this paper, we propose a new method to efficiently compute a representation of an orthogonal basis of the nullspace of a sparse matrix operator BT with B ∈ Rn × m, n > m. We assume that B has full rank, i.e., rank(B) = m. It is well-known that the last n - m columns of the orthogonal matrix Q in a QR factorization B = QR form such a desired null basis. The orthogonal matrix Q can be represented either explicitly as a matrix, or implicitly as a matrix H of Householder vectors. Typically, the matrix H represents the orthogonal factor much more compactly than Q. We will employ this observation to design an efficient block algorithm that computes a sparse representation of the nullspace basis in almost optimal complexity. This new algorithm may, e.g., be used to construct a null space basis of the discrete divergence operator in the finite element context, and we will provide numerical results for this particular application.Publicationtype: Journal Article38 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: without files , Joint domain-decomposition H-LU preconditioners for saddle point problemsFor saddle point problems in fluid dynamics, several popular preconditioners exploit the block structure of the problem to construct block triangular preconditioners. The performance of such preconditioners depends on whether fast, approximate solvers for the linear systems on the block diagonal (representing convection-diffusion problems) as well as for the Schur complement (in the pressure variables) are available. In this paper, we will introduce a completely different approach in which we ignore this given block structure. We will instead compute an approximate LU-factorization of the complete system matrix using hierarchical matrix techniques. In particular, we will use domain-decomposition clustering with an additional local pivoting strategy to order the complete index set. As a result, we obtain an H-matrix structure in which an H-LU factorization is computed more efficiently and with higher accuracy than for the corresponding block structure based clustering. H-LU preconditioners resulting from the block and joint approaches will be discussed and compared through numerical results.Publicationtype: Journal Article40 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: without files , Construction of a discrete divergence-free basis through orthogonal factorization in ℋ-arithmeticIn computational fluid dynamics, linear constraints on the fluid velocity lead to challenging indefinite linear systems of equations. In this paper, we propose to compute an approximation to the constrained linear space of divergence-free functions using hierarchical matrix techniques. This approach will yield a data-sparse, well-conditioned basis of the desired subspace in almost optimal computational complexity which is confirmed by numerical tests. The novelty of this paper lies in the application of hierarchical matrix techniques to orthogonal factorization as well as the construction of an explicit approximation to the subspace basis.Publicationtype: Journal Article36 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: without files , Hierarchical matrix preconditioners for the Oseen equations(2007-02-06)Hierarchical matrices provide a technique for the data-sparse approximation and matrix arithmetic of large, fully populated matrices. In particular, approximate inverses as well as approximate LU factorizations of finite element stiffness matrices may be computed and stored in nearly optimal complexity. In this paper, we develop efficient ℋ-matrix preconditioners for the Oseen equations. In particular, H-matrices will provide efficient preconditioners for the auxiliary (scalar) discrete convection-diffusion and pressure Schur complement problems. We will provide various numerical tests comparing the resulting preconditioners with each other.Publicationtype: Journal Article67 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: without files , Domain-decomposition Based H-LU Preconditioners(2007-01-01); ; Hierarchical matrices (in short: Hilbert space -matrices) have first been introduced in 1998 [7] and since then have entered into a wide range of applications. They provide a format for the data-sparse representation of fully populated matrices.Publicationtype: Journal Article47 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: without files , 37