a preconditioner for the steady state navier stokes equations we present a new method for solving the sparse linear system of equations arising from the discretization of the linearized steady state navier stokes equations also known as the oseen equations the solver is an iterative method of krylov subspace type for which we devise a preconditioner through a heuristic argument based on the fundamental solution tensor for the oseen operator the preconditioner may also be conceived through a weaker heuristic argument involving differential operators computations indicate that convergence for the preconditioned discrete oseen problem is only mildly dependent on the viscosity inverse reynolds number and most importantly that the number of iterations does not grow as the mesh size is reduced indeed since the preconditioner is motivated through analysis of continuous operators the number of iterations decreases for smaller mesh size which accords with better approximation of these operators

efficient and robust schur complement approximations in the augmented lagrangian preconditioner for the incompressible laminar flows this paper introduces three new schur complement approximations for the augmented lagrangian preconditioner the incompressible navier stokes equations discretized by a stabilized finite element method are utilized to evaluate these new approximations of the schur complement a wide range of numerical experiments in the laminar context determines the most efficient schur complement approximation and investigates the effect of the reynolds number mesh anisotropy and refinement on the optimal choice furthermore the advantage over the traditional schur complement approximation is exhibited

parallel preconditioners for monolithic solution of shear bands shear bands are one of the most fascinating instabilities in metals that occur under high strain rates they describe narrow regions on the order of micron scales where plastic deformations and significant heating are localized which eventually leads to fracture nucleation and failure of the component in this work shear bands are described by a set of four strongly coupled thermo mechanical equations discretized by a mixed finite element formulation a thermo viscoplastic flow rule is used to model the inelastic constitutive law and finite thermal conductivity is prescribed which gives rise to an inherent physical length scale governed by competition of shear heating and thermal diffusion the residual equations are solved monolithically by a newton type method at every time step and have been shown to yield mesh insensitive result the jacobian of the system is sparse and has a nonsymmetric block structure that varies with the different stages of shear bands formation the aim of the current work is to develop robust parallel preconditioners to gmres in order to solve the resulting jacobian systems efficiently the main idea is to design schur complements tailored to the specific block structure of the system and that account for the varying stages of shear bands we develop multipurpose preconditioners that apply to standard irreducible discretizations as well as our recent work on isogeometric discretizations of shear bands the proposed preconditioners are tested on benchmark examples and compared to standard state of practice solvers such as gmres ilu and lu direct solvers nonlinear and linear iterations counts as well as cpu times and computational speedups are reported and it is shown that the proposed preconditioners are robust efficient and outperform traditional state of the art solvers

an accurate and efficient method for the incompressible navier stokes equations using the projection method as a preconditioner the projection method is a widely used fractional step algorithm for solving the incompressible navier stokes equations despite numerous improvements to the methodology however imposing physical boundary conditions with projection based fluid solvers remains difficult and obtaining high order accuracy may not be possible for some choices of boundary conditions in this work we present an unsplit linearly implicit discretization of the incompressible navier stokes equations on a staggered grid along with an efficient solution method for the resulting system of linear equations since our scheme is not a fractional step algorithm it is straightforward to specify general physical boundary conditions accurately however this capability comes at the price of having to solve the time dependent incompressible stokes equations at each timestep to solve this linear system efficiently we employ a krylov subspace method preconditioned by the projection method in our implementation the subdomain solvers required by the projection preconditioner employ the conjugate gradient method with geometric multigrid preconditioning the accuracy of the scheme is demonstrated for several problems including forced and unforced analytic test cases and lid driven cavity flows these tests consider a variety of physical boundary conditions with reynolds numbers ranging from 1 to 30000 the effectiveness of the projection preconditioner is compared to an alternative preconditioning strategy based on an approximation to the schur complement for the time dependent incompressible stokes operator the projection method is found to be a more efficient preconditioner in most cases considered in the present work

a taxonomy and comparison of parallel block multi level preconditioners for the incompressible navier stokes equations in recent years considerable effort has been placed on developing efficient and robust solution algorithms for the incompressible navier stokes equations based on preconditioned krylov methods these include physics based methods such as simple and purely algebraic preconditioners based on the approximation of the schur complement all these techniques can be represented as approximate block factorization abf type preconditioners the goal is to decompose the application of the preconditioner into simplified sub systems in which scalable multi level type solvers can be applied in this paper we develop a taxonomy of these ideas based on an adaptation of a generalized approximate factorization of the navier stokes system first presented in a quarteroni f saleri a veneziani factorization methods for the numerical approximation of navier stokes equations computational methods in applied mechanical engineering 188 2000 505 526 this taxonomy illuminates the similarities and differences among these preconditioners and the central role played by efficient approximation of certain schur complement operators we then present a parallel computational study that examines the performance of these methods and compares them to an additive schwarz domain decomposition dd algorithm results are presented for two and three dimensional steady state problems for enclosed domains and inflow outflow systems on both structured and unstructured meshes the numerical experiments are performed using mpsalsa a stabilized finite element code

pressure correction algebraic splitting methods for the incompressible navier stokes equations in this paper we present a new family of methods for the effective numerical solution of the incompressible unsteady navier stokes equations these methods resort to an algebraic splitting of the discretized problem based on inexact lu block factorizations of the corresponding matrix following a quarteroni f saleri and a veneziani comput methods appl mech engrg 188 2000 pp 505 526 in particular we will start from inexact algebraic factorizations of algebraic chorin temam and yosida type and introduce a pressure correction step aimed at improving the time accuracy one of the schemes obtained in this way the algebraic chorin temam pressure correction method resembles a method previously introduced in the framework of differential projection schemes see l timmermans p minev and f v de vosse internat j numer methods fluids 22 1996 pp 673 688 a prohl projection and quasi compressibility methods for solving the incompressible navier stokes equations teubner stuttgart 1997 the stability and the dependence of splitting error on the time step of the new methods is investigated and tested on several numerical cases

development of a stokes flow solver robust to large viscosity jumps using a schur complement approach with mixed precision arithmetic we develop an iterative solution technique for solving stokes flow problems with smooth and discontinuous viscosity structures using a three dimensional staggered grid finite difference discretization two preconditioned iterative methodologies are applied to the saddle point arising from the discrete stokes problem they consist of a velocity pressure coupled approach fc and a decoupled schur complement approach sc within both of these methods we utilize either the scaled bfbt or an identity matrix scaled by the local cell viscosity lv to define a preconditioner for the schur complement additionally we propose to use a mixed precision krylov kernel to improve the convergence by reducing round off error in this approach standard double precision is used during the application of the preconditioner whilst higher precision arithmetic is used to define the matrix vector product dot products and norms required by the krylov method in our krylov kernel we utilize quad precision arithmetic which is emulated via the double double precision method we consider several simplified geodynamic problems with a viscosity contrast to demonstrate the robustness and scalability of our solution methods through a careful choice of stopping conditions we are able to quantitatively compare the residuals between the sc and fc approaches we examine the trade off relationship between the number of outer iterations required for convergence and the computational cost per iteration for the each solution methods we find that it is advantageous to use the fc approach utilizing relaxed tolerances for solution of the sub problems combined with the lv preconditioner we also observed that in general the sc approach is more robust than fc and that bfbt is more robust than lv when used in our numerical experimental in addition our mixed precision method produces improved convergence rates of arnoldi type krylov subspace methods without a drastic increasing the computational time the usage of a high precision krylov kernel is found to be useful for the solver associated with the velocity sub problem

a note on preconditioning for indefinite linear systems preconditioners are often conceived as approximate inverses for nonsingular indefinite matrices of saddle point or kkt form we show how preconditioners incorporating an exact schur complement lead to preconditioned matrices with exactly two or exactly three distinct eigenvalues thus approximations of the schur complement lead to preconditioners which can be very effective even though they are in no sense approximate inverses

a priori sparsity patterns for parallel sparse approximate inverse preconditioners parallel algorithms for computing sparse approximations to the inverse of a sparse matrix either use a prescribed sparsity pattern for the approximate inverse or attempt to generate a good pattern as part of the algorithm this paper demonstrates that for pde problems the patterns of powers of sparsified matrices psms can be used a priori as effective approximate inverse patterns and that the additional effort of adaptive sparsity pattern calculations may not be required psm patterns are related to various other approximate inverse sparsity patterns through matrix graph theory and heuristics concerning the pde s green s function a parallel implementation shows that psm patterned approximate inverses are significantly faster to construct than approximate inverses constructed adaptively while often giving preconditioners of comparable quality

preconditioning for the steady state navier stokes equations with low viscosity we introduce a preconditioner for the linearized navier stokes equations that is effective when either the discretization mesh size or the viscosity approaches zero for constant coefficient problems with periodic boundary conditions we show that the preconditioning yields a system with a single eigenvalue equal to 1 so that performance is independent of both viscosity and mesh size for other boundary conditions we demonstrate empirically that convergence depends only mildly on these parameters and we give a partial analysis of this phenomenon we also show that some expensive subsidiary computations required by the new method can be replaced by inexpensive approximate versions of these tasks based on iteration with virtually no degradation of performance

block preconditioners based on approximate commutators this paper introduces a strategy for automatically generating a block preconditioner for solving the incompressible navier stokes equations we consider the pressure convection diffusion preconditioners proposed by kay loghin and wathen siam j sci comput 24 2002 pp 237 256 and silvester elman kay and wathen j comput appl math 128 2001 pp 261 279 numerous theoretical and numerical studies have demonstrated mesh independent convergence on several problems and the overall efficacy of this methodology a drawback however is that it requires the construction of a convection diffusion operator denoted f p projected onto the discrete pressure space this means that integration of this idea into a code that models incompressible flow requires a sophisticated understanding of the discretization and other implementation issues something often held only by the developers of the model as an alternative we consider automatic ways of computing f p based on purely algebraic considerations the new methods are closely related to the bfbt preconditioner of elman siam j sci comput 20 1999 pp 1299 1316 we use the fact that the preconditioner is derived from considerations of commutativity between the gradient and convection diffusion operators together with methods for computing sparse approximate inverses to generate the required matrix f p automatically we demonstrate that with this strategy the favorable convergence properties of the preconditioning methodology are retained

efficient preconditioning of the linearized navier stokes equations for incompressible flow we outline a new class of robust and efficient methods for solving subproblems that arise in the linearization and operator splitting of navier stokes equations we describe a very general strategy for preconditioning that has two basic building blocks a multigrid v cycle for the scalar convection diffusion operator and a multigrid v cycle for a pressure poisson operator we present numerical experiments illustrating that a simple implementation of our approach leads to an effective and robust solver strategy in that the convergence rate is independent of the grid robust with respect to the time step and only deteriorates very slowly as the reynolds number is increased

field of values analysis of preconditioned linearized rayleigh benard convection problems abstract in this paper we use the notion of field of values fov equivalence of matrices to study a class of block triangular preconditioners for the fixed point linearization of the rayleigh benard convection problem discretized with inf sup stable finite element spaces first sufficient conditions on the nondimensional parameters of the problem are determined in order to establish the fov equivalence between the system matrix and the preconditioners four upper triangular block preconditioners belonging to the general proposed class are then considered numerical experiments show that the generalized minimal residual gmres convergence is robust with respect to the mesh size for these preconditioned systems we also compare the performance of the different preconditioners in terms of computational time

preconditioning techniques for large linear systems a survey this article surveys preconditioning techniques for the iterative solution of large linear systems with a focus on algebraic methods suitable for general sparse matrices covered topics include progress in incomplete factorization methods sparse approximate inverses reorderings parallelization issues and block and multilevel extensions some of the challenges ahead are also discussed an extensive bibliography completes the paper

skeletonization of ribbon like shapes based on a new wavelet function a wavelet based scheme to extract skeleton of ribbon like shape is proposed in this paper where a novel wavelet function plays a key role in this scheme which possesses three significant characteristics namely 1 the position of the local maximum moduli of the wavelet transform with respect to the ribbon like shape is independent of the gray levels of the image 2 when the appropriate scale of the wavelet transform is selected the local maximum moduli of the wavelet transform of the ribbon like shape produce two new parallel contours which are located symmetrically at two sides of the original one and have the same topological and geometric properties as that of the original shape 3 the distance between these two parallel contours equals to the scale of the wavelet transform which is independent of the width of the shape this new scheme consists of two phases 1 generation of wavelet skeleton based on the desirable properties of the new wavelet function symmetry analyses of the maximum moduli of the wavelet transform is described midpoints of all pairs of contour elements can be connected to generate a skeleton of the shape which is defined as wavelet skeleton 2 modification of the wavelet skeleton thereafter a set of techniques are utilized for modifying the artifacts of the primary wavelet skeleton the corresponding algorithm is also developed in this paper experimental results show that the proposed scheme is capable of extracting exactly the skeleton of the ribbon like shape with different width as well as different gray levels the skeleton representation is robust against noise and affine transformation

an analysis of the fractional step method the fractional step method for solving the incompressible navier stokes equations in primitive variables is analyzed as a block lu decomposition in this formulation the issues involving boundary conditions for the intermediate velocity variables and the pressure are clearly resolved in addition it is shown that poor temporal accuracy first order is not due to boundary conditions but due to the method itself a generalized block lu decomposition that overcomes this difficulty is presented allowing arbitrarily high temporal order of accuracy the generalized decomposition is shown to be useful for a wide range of problems including steady problems technical issues such as stability and the appropriate pressure update scheme are also addressed numerical simulations of the steady incompressible navier stokes equations in a square domain confirm the theoretical results 18 refs 4 figs

a variant of relaxed triangular splitting preconditioners for generalized saddle point problems from navier stokes equations abstract based on the dimensional split preconditioner in this paper a variant of relaxed triangular splitting preconditioner vrts is presented and discussed in which a simple and feasible way is designed for the selection of the parameters spectral properties of the vrts preconditioned matrix are analyzed in detail theoretical analysis shows that when applying the vrts preconditioner within a krylov subspace method less computational cost is required at each iteration than some state of the art approaches finally some numerical experiments arising from the discretization of navier stokes equations are given to illustrate and validate the efficiency and robustness of the presented preconditioners with using gmres as an iterative solver

an extended finite element method applied to levitated droplet problems we consider a standard model for incompressible two phase flows in which a localized force at the interface describes the effect of surface tension if a level set method is applied then the approximation of the interface is in general not aligned with the triangulation this causes severe difficulties w r t the discretization and often results in large spurious velocities in this paper we reconsider a modified extended finite element method xfem which in previous papers has been investigated for relatively simple two phase flow model problems and apply it to a physically realistic levitated droplet problem the results show that due to the extension of the standard fe space one obtains much better results in particular for large interface tension coefficients furthermore a certain cut off technique results in better efficiency without sacrificing accuracy copyright 2010 john wiley sons ltd

a hierarchical low rank schur complement preconditioner for indefinite linear systems nonsymmetric and highly indefinite linear systems can be quite difficult to solve by iterative methods this paper combines ideas from the multilevel schur low rank preconditioner developed by y x

facsi a block parallel preconditioner for fluid structure interaction in hemodynamics modeling fluid structure interaction fsi in the vascular system is mandatory to reliably compute mechanical indicators in vessels undergoing large deformations in order to cope with the computational complexity of the coupled 3d fsi problem after discretizations in space and time a parallel solution is often mandatory in this paper we propose a new block parallel preconditioner for the coupled linearized fsi system obtained after space and time discretization we name it facsi to indicate that it exploits the factorized form of the linearized fsi matrix the use of static condensation to formally eliminate the interface degrees of freedom of the fluid equations and the use of a simple preconditioner for saddle point problems facsi is built upon a block gauss seidel factorization of the fsi jacobian matrix and it uses ad hoc preconditioners for each physical component of the coupled problem namely the fluid the structure and the geometry in the fluid subproblem after operating static condensation of the interface fluid variables we use a simple preconditioner on the reduced fluid matrix moreover to efficiently deal with a large number of processes facsi exploits efficient single field preconditioners e g based on domain decomposition or the multigrid method we measure the parallel performances of facsi on a benchmark cylindrical geometry and on a problem of physiological interest namely the blood flow through a patient specific femoropopliteal bypass we analyze the dependence of the number of linear solver iterations on the cores count scalability of the preconditioner and on the mesh size optimality c 2016 elsevier inc all rights reserved

robust preconditioners for incompressible mhd models in this paper we develop two classes of robust preconditioners for the structure preserving discretization of the incompressible magnetohydrodynamics mhd system by studying the well posedness of the discrete system we design block preconditioners for them and carry out rigorous analysis on their performance we prove that such preconditioners are robust with respect to most physical and discretization parameters in our proof we improve the existing estimates of the block triangular preconditioners for saddle point problems by removing the scaling parameters which are usually difficult to choose in practice this new technique is applicable not only to the mhd system but also to other problems moreover we prove that krylov iterative methods with our preconditioners preserve the divergence free condition exactly which complements the structure preserving discretization another feature is that we can directly generalize this technique to other discretizations of the mhd system we also present preliminary numerical results to support the theoretical results and demonstrate the robustness of the proposed preconditioners

parallel preconditioning with sparse approximate inverses a parallel preconditioner is presented for the solution of general sparse linear systems of equations a sparse approximate inverse is computed explicitly and then applied as a preconditioner to an iterative method the computation of the preconditioner is inherently parallel and its application only requires a matrix vector product the sparsity pattern of the approximate inverse is not imposed a priori but captured automatically this keeps the amount of work and the number of nonzero entries in the preconditioner to a minimum rigorous bounds on the clustering of the eigenvalues and the singular values are derived for the preconditioned system and the proximity of the approximate to the true inverse is estimated an extensive set of test problems from scientific and industrial applications provides convincing evidence of the effectiveness of this approach

pressure schur complement preconditioners for the discrete oseen problem we consider several preconditioners for the pressure schur complement of the discrete steady oseen problem two of the preconditioners are well known from the literature and the other is new supplemented with an appropriate approximate solve for an auxiliary velocity subproblem these approaches give rise to a family of the block preconditioners for the matrix of the discrete oseen system in the paper we critically review possible advantages and difficulties of using various schur complement preconditioners we recall existing eigenvalue bounds for the preconditioned schur complement and prove such for the newly proposed preconditioner these bounds hold both for lbb stable and stabilized finite elements results of numerical experiments for several model two dimensional and three dimensional problems are presented in the experiments we use lbb stable finite element methods on uniform triangular and tetrahedral meshes one particular conclusion is that in spite of essential improvement in comparison with simple scaled mass matrix preconditioners in certain cases none of the considered approaches provides satisfactory convergence rates in the case of small viscosity coefficients and a sufficiently complex e g circulating advection vector field

multi space reduced basis preconditioners for parametrized stokes equations abstract we introduce a two level preconditioner for the efficient solution of large scale saddle point linear systems arising from the finite element fe discretization of parametrized stokes equations this preconditioner extends the multi space reduced basis msrb preconditioning method proposed in dal santo et al 2018 it combines an approximated block fine grid preconditioner with a reduced basis rb solver which plays the role of coarse component a sequence of rb spaces constructed either with an enriched velocity formulation or a petrov galerkin projection is built each rb coarse component is defined to perform a single iteration of the iterative method at hand the flexible gmres fgmres algorithm is employed to solve the resulting preconditioned system and targets small tolerances with a very small iteration count and in a very short time numerical test cases for stokes flows in three dimensional parameter dependent geometries are considered to assess the numerical properties of the proposed technique in different large scale computational settings