Schur complement preconditioners for the Navier–Stokes equations

In this paper, first, we use the Rich type iterative method for numerical solution of nonlinear fuzzy Fredholm integral equation. The main structure of this method is based on the Richardson iterative method and trapezoidal quadrature formula for Numerical solution of nonlinear fuzzy integral equations. Convergence, error analysis and numerical stability of the proposed method are also discussed.

This paper is a matrix iterative method presented to compute the solutions of the matrix equation, AXB=C, with unknown matrix X∈𝒮, where 𝒮 is the constrained matrices set like symmetric, symmetric-R-symmetric and (R, S)-symmetric. By this iterative method, for any initial matrix X 0∈𝒮, a solution X* can be obtained within finite iteration steps if exact arithmetics were used, and the solution X* with the minimum Frobenius norm can be obtained by choosing a special kind of initial matrix. The solution , which is nearest to a given matrix ˜X in Frobenius norm, can be obtained by first finding the minimum Frobenius norm solution of a new compatible matrix equation. The numerical examples given here show that the iterative method proposed in this paper has faster convergence and higher accuracy than the iterative methods proposed in [G.-X. Huang, F. Yin, and K. Guo, An iterative method for the skew-symmetric solution and the optimal approximate solution of the matrix equation AXB=C , J. Comput. Appl. Math. 212 (2008), pp. 231–244; Y. Lei and A.-P. Liao, A minimal residual algorithm for the inconsistent matrix equation AXB=C over symmetric matrices, Appl. Math. Comput. 188 (2007), pp. 499–513; Z.-Y. Peng, An iterative method for the least squares symmetric solution of the linear matrix equation AXB=C , Appl. Math. Comput. 170 (2005), pp. 711–723; Y.-X. Peng, X.-Y. Hu, and L. Zhang, An iteration method for the symmetric solutions and the optimal approximation solution of the matrix equation AXB=C , Appl. Math. Comput. 160 (2005), pp. 763–777].

We present in this paper a method for the numerical solution of the steady and unsteady Navier-Stokes equations for incompressible viscous fluids. This method is based on the following techniques: • A mixed finite element approximation acting on a pressure-velocity formulation of the problem, • A time discretization by finite differences for the unsteady problem, • An iterative method using — via a convenient nonlinear least square formulation — a conjugate gradient algorithm with scaling; the scaling makes a fundamental use of an efficient Stokes solver associated to the above mixed finite element approximation.

HiFlow3 Version 2.0 continues the path as a multi-purpose finite element software, which provides powerful tools for efficient and accurate solution of a wide range of problems modeled by partial differen- tial equations (PDEs). New features and functionalities, which allow to run numerical simulations with more advanced solution algorithms and discretizations in comparison to previous releases of HiFlow3, have been implemented. These comprise Uncertainty Quantification (UQ), energy-efficient multigrid techniques, Schur complement solvers for saddle-point problems, extended third-party library support, and adaptive local mesh refinement in a parallel computing environment. Furthermore, HiFlow3 has been successfully integrated into advanced and state-of-the-art simulation environments by means of the Medical Simulation Markup Language (MSML), for example.The presented new algorithms and features as well as general under-the-hood improvements have enabled excellent and relevant research activities in the fields of both medical engineering and meteorol- ogy and environmental sciences. The described show cases demonstrate the potential and advantages, which HiFlow3 can offer in performing a numerical simulation by means of finite element methods. Especially, the high performance computing capabilities of HiFlow3 – not only in the mentioned fields of applications, but also in general – have been significantly improved in Version 2.0.

In literature, it has been reported that the convergence of some preconditioned stationary iterative methods using certain type upper triangular matrices as preconditioners are faster than the basic iterative methods. In this paper, a new preconditioned iterative method for the numerical solution of linear systems has been introduced, and the convergence analysis of the proposed method and an existing one have been done. Some numerical examples have also been given, which show the effectiveness of both of the methods.

The simulation of several physical phenomena, arising from a wide class of scientific fields, requires solving large sparse linear systems effectively. In order to exploit appropriately the available supercomputing infrastructures, designing efficient and scalable solvers for large sparse linear systems is necessary. A hybrid parallel algebraic iterative method based on the Schur complement method and multiprojection techniques that utilizes semiaggregated subdomains is proposed. The graph corresponding to the coefficient matrix of the sparse linear system is partitioned by a graph partitioning algorithm and the Schur complement problem is solved by a preconditioned Krylov subspace method. The preconditioning scheme is based on domain decomposition and the resulting subdomains consist of fine and aggregated components from the inner boundaries. Moreover, two variants for approximating the local Schur complements, derived from the semiaggregation procedure, are proposed in order to improve memory requirements and preprocessing time. The proposed method is suitable for distributed memory systems with multicore nodes and has improved convergence behavior for a large number of workstations. Numerical and comparative results concerning the applicability, the convergence behavior, and the parallel performance of the proposed scheme are given.

This study presents an efficacy comparison of iterative solution methods for solving the compressible-fluid Reynolds equation in modeling air- or gas-lubricated bearings. A direct fixed-point iterative (DFI) method and Newton’s method are employed to transform the Reynolds equation in a form that can be solved iteratively. The iterative solution methods examined are the Gauss–Seidel method, the successive over-relaxation (SOR) method, the preconditioned conjugate gradient (PCG) method, and the multigrid method. The overall solution time is affected by both the transformation method and the iterative method applied. In this study, Newton’s method shows its effectiveness over the straightforward DFI method when the same iterative method is used. It is demonstrated that the use of an optimal relaxation factor is of vital importance for the efficiency of the SOR method. The multigrid method is an order faster than the PCG and optimal SOR methods. Also, the multigrid and PCG methods involve an extended coding work and are less flexible in dealing with gridwork and boundary conditions. Consequently, a compromise has to be made in terms of ease of use as well as programming effort for the solution of the compressible-fluid Reynolds equation.

This chapter discusses iterative methods for the solution of very large severely ill-conditioned linear systems of equations that arise from the discretization of linear ill-posed problems. The right-hand side vector represents the given data and is assumed to be contaminated by errors. Solution methods proposed in the literature employ some form of filtering to reduce the influence of the error in the right-hand side on the computed approximate solution. The amount of filtering is determined by a parameter often referred to as the regularization parameter. We discuss how the filtering affects the computed approximate solution and consider the selection of regularization parameter. Methods in which a suitable value of the regularization parameter is determined during the computation without user intervention are emphasized. New iterative solution methods based on expanding explicitly chosen filter functions in terms of Chebyshev polynomials are presented. The properties of these methods are illustrated with applications to image restoration.

Development of a Stokes flow solver robust to large viscosity jumps using a Schur complement approach with mixed precision arithmetic

The execution of iteration methods, including iteration methods for time-dependent differential equations, is one of the most time-consuming tasks in scientific computing, especially when large systems have to be solved, large time scales have to be computed or solutions of high quality are required. To decrease the computation time, parallel machines can be used but a detailed analysis of the potential parallelism is needed for an efficient implementation as data dependencies can strongly limit the performance. This article describes and compares the potential parallelism of various iterative solution methods for linear and non-linear equation systems and ordinary differential equations. Methods that have been especially designed for parallel execution as well as parallel implementations of modern efficient sequential solution methods are considered.

A matching Schur complement preconditioning technique for inverse source problems

Element-by-element iterative solution methods have been shown to be effective for solving large, sparse linear systems of equations with a positive definite stiffness matrix. In this paper the iterative method GMRES will be described. This solution method is also suited for solving nonsymmetric systems of equations. As acceleration technique an element-by-element preconditioner has been employed. Attention is focused mainly on the convergence behaviour of the method. The calculations described in this paper have been performed with the finite element package DIANA.

This paper presents a general theory of maximally orthogonalized div- and curl-conforming higher order basis functions (HOBFs) over generalized wires, quadrilaterals, and hexahedra. In particular, all elements of such bases, necessary for fast and easy implementation, are listed up to order <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> =8. Numerical results, given for div-conforming bases applied in an iterative method of moments solution of integral equations, show that the condition number and the number of iterations are a) much lower than in the case of other HOBFs of polynomial type and b) practically not dependent on the applied expansion order.

Geophysical Inverse Theory and Regularization Problems

This paper is devoted to develop a robust numerical method to solve a system of complementarity problems arising from pricing American options under regime switching. Based on a penalty method, the system of complementarity problems are approximated by a set of coupled nonlinear partial differential equations (PDEs). We then introduce a fitted finite volume method for the spatial discretization along with a fully implicit time stepping scheme for the PDEs, which results in a system of nonlinear algebraic equations. We show that this scheme is consistent, stable and monotone, hence convergent. To solve the system of nonlinear equations effectively, an iterative solution method is established. The convergence of the solution method is shown. Numerical tests are performed to examine the convergence rate and verify the effectiveness and robustness of the new numerical scheme.