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

variants of the deteriorated pss preconditioner for saddle point problems two new preconditioners which can be viewed as variants of the deteriorated positive definite and skew hermitian splitting preconditioner are proposed for solving saddle point problems the corresponding iteration methods are proved to be convergent unconditionally for cases with positive definite leading blocks the choice strategies of optimal parameters for the two iteration methods are discussed based on two recent optimization results for extrapolated cayley transform which result in faster convergence rate and more clustered spectrum compared with some preconditioners of similar structures the new preconditioners have better convergence properties and spectrum distributions in addition more practical preconditioning variants of the new preconditioners are considered numerical experiments are presented to illustrate the advantages of the new preconditioners over some similar preconditioners to accelerate gmres c 2018 elsevier ltd all rights reserved

saving comparisons in the crochemore perrin string matching algorithm crochemore and perrin discovered an elegant linear time constant space string matching algorithm that makes at most 2n m symbol comparisons this paper shows how to modify their algorithm to use fewer comparisons given any fixed e 0 the new algorithm takes linear time uses constant space and makes at most n 1 e2 n m symbol comparisons if o log m space is available then the algorithm makes at most n 12 n m symbol comparisons the pattern preprocessing step also takes linear time and uses constant space these are the first string matching algorithms that make fewer than 2n m symbol comparisons and use sub linear space

a block preconditioner for non isothermal flow in porous media abstract in petroleum reservoir simulation the industry standard preconditioner the constrained pressure residual method cpr is a two stage process which involves solving a restricted pressure system with algebraic multigrid amg initially designed for isothermal models this approach is often used in the thermal case however it does not have a specific treatment of the additional energy conservation equation and temperature variable we seek to develop preconditioners which better capture thermal effects such as heat diffusion in order to study the effects of both pressure and temperature on fluid and heat flow we consider a model of non isothermal single phase flow through porous media for this model we develop a block preconditioner with an efficient schur complement approximation both the pressure block and the approximate schur complement are approximately inverted using an amg v cycle the resulting solver is scalable with respect to problem size and parallelization

fast iterative solution of stabilised stokes systems part i using simple diagonal preconditioners mixed finite element approximation of the classical stokes problem describing slow viscous incompressible flow gives rise to symmetric indefinite systems for the discrete velocity and pressure variables iterative solution of such indefinite systems is feasible and is an attractive approach for large problems the use of stabilisation methods for convenient but unstable mixed elements introduces stabilisation parameters we show how these can be chosen to obtain rapid iterative convergence we propose a conjugate gradient like method the method of preconditioned conjugate residuals which is applicable to symmetric indefinite problems describe the effects of stabilisation on the algebraic structure of the discrete stokes operator and derive estimates of the eigenvalue spectrum of this operator on which the convergence rate of the iteration depends here we discuss the simple case of diagonal preconditioning our results apply to both locally and globally stabilised mixed elements as well as to elements which are inherently stable we demonstrate that convergence rates comparable to that achieved using the diagonally scaled conjugate gradient method applied to the discrete laplacian are approachable for the stokes problem

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

high order methods for the approximation of the incompressible navier stokes equations in a moving domain in this paper we address the numerical approximation of the incompressible navier stokes equations in a moving domain by the spectral element method and high order time integrators we present the arbitrary lagrangian eulerian ale formulation of the incompressible navier stokes equations and propose a numerical method based on the following kernels a lagrange basis associated with fekete points in the spectral element method context bdf time integrators an ale map of high degree and an algebraic linear solver in particular the high degree ale map is appropriate to deal with a computational domain whose boundary is described with curved elements finally we apply the proposed strategy to a test case c 2011 elsevier b v all rights reserved

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

homology of commutative algebras and an invariant of simis and vasconcelos abstract simis and vasconcelos 8 and 9 have introduced for any ideal i of finite type in a commutative ring r an associated module i which is an invariant ofi i measures the extent to which the koszul complex of a set of generators ofi gives one information on the syzygies of the conormal modulei i2 in this paper we show that i is isomorphic toh2 r r i r i the second andre quillen homology of ther algebra r i see 1 one can obtain special cases of this result by combining alternative invariant formulations of i from 9 with results of andre however the general case seems to be most easily proved by using crossed module techniques adapted from ideas of lichtenbaum and schlessinger 4 and from the corresponding theory in the category of groups the only relatively deep results from the homology of commutative algebras that we shall use are necessary to identify the lichtenbaum schlessinger homology group t2 with the corresponding andre quillen homology h2 a reference for this result is 1 p 206 proposition 12

torsion freeness of symmetric powers of ideals let i be an ideal in a noetherian commutative ring r with unit let k 2 be an integer and let k i s k i i k be the canonical surjective r module homomorphism from the kth symmetric power of i to the kth power of i when pd r i 1 or when i is a perfect gorenstein ideal of grade 3 we provide a necessary and sufficient condition for k to be an isomorphism in terms of upper bounds for the minimal number of generators of the localisations of i when i m is a maximal ideal of r we show that is an isomorphism if and only if r m is a regular local ring in all three cases for i our results yield that if is an isomorphism then at is also an isomorphism for each 1 t k

eigenvalue bounds for the schur complement with a pressure convection diffusion preconditioner in incompressible flow computations if the stationary navier stokes system or an implicit time discretization of the evolutionary navier stokes system is linearized by a picard iteration and discretized in space by a mixed finite element method there arises a saddle point system which may be solved by a krylov subspace method or an uzawa type approach for each of these resolution methods it is necessary to precondition the schur complement associated to the saddle point problem in question in the work at hand we give upper and lower bounds of the eigenvalues of this schur complement under the assumption that it is preconditioned by a pressure convection diffusion matrix

a multigrid preconditioner for stabilised discretisations of advection diffusion problems abstract this paper is concerned with the iterative solution of linear systems arising from stabilised discretisations of advection diffusion problems on stretched finite element grids using nonuniform grids of this type leads in general to very badly conditioned matrix problems we therefore consider using gmres in conjunction with a multigrid mg preconditioning strategy in particular we show that in order to achieve the grid size independent convergence which is characteristic of mg methods it is essential to use an effective stabilisation strategy at each level of the mg structure

modeling studies and efficient numerical methods for proton exchange membrane fuel cell abstract in this paper a three dimensional 3d nonisothermal multiphysics two phase steady state transport model and its efficient numerical methods are systematically studied for a full proton exchange membrane fuel cell pemfc in the sense of efficiency and accuracy the conservation equations of mass momentum species charge and energy are fully addressed in view of nonisothermality and multiphase characteristics in addition from an accurate numerical discretization s point of view we present some new formulations for species equations by investigating the interactions among the species in a framework of the combined finite element upwind finite volume method some efficient numerical methods are developed in terms of kirchhoff transformation for the sake of a fast and convergent numerical simulation the 3d simulations demonstrate that the convergent solutions can be attained within 80 nonlinear iterations in contrast to the oscillating and nonconvergent iterations conducted by commercial flow solvers or in house code with standard finite element volume methods numerical convergence tests are carried out to verify the efficiency and accuracy of our numerical algorithms and techniques

multigrid solution of the incompressible navier stokes equations in general coordinates galerkin coarse grid approximation gca in multigrid methods is investigated for the incompressible navier stokes equations in general coordinates an efficient algorithm performing gca is presented the behavior of coarse grid matrices is studied under gca with different transfer operators for square and l shaped driven cavity problems the performance of the multigrid method using different combinations of transfer operators for the computation of coarse grid matrices and of coarse grid correction is investigated further computations are carried out in general coordinates for a channel flow problem with backward facing step in three dimensions

on parameter choice and iterative convergence for stabilised discretisations of advection diffusion problems abstract in this work we consider the design of robust and efficient finite element approximation methods for solving advection diffusion equations specifically we consider the stabilisation of discrete approximations using uniform grids which do not resolve boundary layers as might arise using a multi level or multigrid iteration strategy to solve the discrete problem our analysis shows that when using supg streamline upwind finite element methodology there is a symbiotic relationship between best solution approximation and fast convergence of smoothers based on the standard gmres iteration we also show that stabilisation based on simple artificial diffusion perturbation terms an approach often advocated by multigrid practitioners is less appealing

an augmented lagrangian based approach to the oseen problem we describe an effective solver for the discrete oseen problem based on an augmented lagrangian formulation of the corresponding saddle point system the proposed method is a block triangular preconditioner used with a krylov subspace iteration like bicgstab the crucial ingredient is a novel multigrid approach for the 1 1 block which extends a technique introduced by schoberl for elasticity problems to nonsymmetric problems our analysis indicates that this approach results in fast convergence independent of the mesh size and largely insensitive to the viscosity we present experimental evidence for both isop2 p0 and isop2 p1 finite elements in support of our conclusions we also show results of a comparison with two state of the art preconditioners showing the competitiveness of our approach

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 note on preconditioning nonsymmetric matrices the preconditioners for indefinite matrices of kkt form in m f murphy g h golub and a j wathen siam j sci comput 21 2000 pp 1969 1972 are extended to general nonsymmetric matrices

discrete interpolation norms with applications we describe norm representations for interpolation spaces generated by finite dimensional subspaces of hilbert spaces these norms are products of integer and noninteger powers of the grammian matrices associated with the generating pair of spaces for the interpolation space we include a brief description of some of the algorithms which allow the efficient computation of matrix powers we consider in some detail the case of fractional sobolev spaces both for positive and negative indices together with applications arising in preconditioning techniques numerical experiments are included

an iteration for indefinite systems and its application to the navier stokes equations for large sparse systems of linear equations iterative solution techniques are attractive in this paper we propose and examine the convergence of an iterative method for an important class of nonsymmetric and indefinite coefficient matrices based on the use of an indefinite and symmetric preconditioner we apply our technique to the linearized navier stokes equations the oseen equations

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

fast iterative solution of stabilised stokes systems part ii using general block preconditioners mixed finite element approximation of the classical stokes problem describing slow viscous incompressible flow gives rise to symmetric indefinite systems for the discrete velocity and pressure variables iterative solution of such indefinite systems is feasible and is an attractive approach for large problems part i of this work described a conjugate gradient like method the method of preconditioned conjugate residuals which is applicable to symmetric indefinite problems a j wathen and d j silvester siam j numer anal 30 1993 pp 630 649 using simple arguments estimates for the eigenvalue distribution of the discrete stokes operator on which the convergence rate of the iteration depends are easily derived part i discussed the important case of diagonal preconditioning scaling this paper considers the more general class of block preconditioners where the partitioning into blocks corresponds to the natural partitioning into the velocity and pressure variables it is shown that provid

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

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

stabilised vs stable mixed methods for incompressible flow abstract the accuracy of low order mixed finite element methods for the incompressible navier stokes equations is investigated in this work some numerical experiments suggest that the lowest order stabilised p 1 p 0 method linear velocity constant pressure is more efficient than the alternative a priori stable non conforming crouzeix raviart p 1 p 0 approach the relative accuracy of stabilised p 1 p 0 and p 1 p 1 linear velocity continuous linear pressure is assessed herein

analysis of iterative methods for the steady and unsteady stokes problem application to spectral element discretizations a new and detailed analysis of the basic uzawa algorithm for decoupling of the pressure and the velocity in the steady and unsteady stokes operator is presented the paper focuses on the following new aspects explicit construction of the uzawa pressure operator spectrum for a semiperiodic model problem general relationship of the convergence rate of the uzawa procedure to classical inf sup discretization analysis and application of the method to high order variational discretization

numerical approximation of phase field based shape and topology optimization for fluids we consider the problem of finding optimal shapes of fluid domains the fluid obeys the navier stokes equations inside a holdall container we use a phase field approach using diffuse interfaces to describe the domain of free flow we formulate a corresponding optimization problem where flow outside the fluid domain is penalized the resulting formulation of the shape optimization problem is shown to be well posed hence there exists a minimizer and first order optimality conditions are derived for the numerical realization we introduce a mass conserving gradient flow and obtain a cahn hilliard type system which is integrated numerically using the finite element method an adaptive concept using reliable residual based error estimation is exploited for the resolution of the spatial mesh the overall concept is numerically investigated and comparison values are provided

fast nonsymmetric iterations and preconditioning for navier stokes equations discretization and linearization of the steady state navier stokes equations gives rise to a nonsymmetric indefinite linear system of equations in this paper we introduce preconditioning techniques for such systems with the property that the eigenvalues of the preconditioned matrices are bounded independently of the mesh size used in the discretization we confirm and supplement these analytic results with a series of numerical experiments indicating that krylov subspace iterative methods for nonsymmetric systems display rates of convergence that are independent of the mesh parameter in addition we show that preconditioning costs can be kept small by using iterative methods for some intermediate steps performed by the preconditioner

a block preconditioner for high order mixed finite element approximations to the navier stokes equations an iterative solver for the linear system arising from a discretization by high order mixed finite elements for the linearized steady state navier stokes problem is presented two mixed finite elements are considered mathbf p k 1 p k and mathbf p k 2 p k the iterative solver is based on the gmres method with right preconditioning where the preconditioner has a block upper triangular form computational results show that the convergence rate seems to be independent of the grid size parameter h and the polynomial degree k but is slightly affected by the increasing of the reynolds number

combining the augmented lagrangian preconditioner with the simple schur complement approximation the augmented lagrangian al preconditioner and its variants have been successfully applied to solve saddle point systems arising from the incompressible navier stokes equations discretized by th

on computational issues of immersed finite element methods the objective of this paper is to provide a review of recent finite element formulations for immersed methods in these finite element formulations independent lagrangian solid meshes are introduced to move on top of a background eulerian fluid mesh this key feature allows the handling without excessive fluid mesh adaptation multiple deformable solids immersed in viscous fluid in particular pros and cons of both explicit and implicit approaches are illustrated along with subtle differences between incompressible and slightly compressible models

augmented lagrangian preconditioner for large scale hydrodynamic stability analysis abstract hydrodynamic linear stability analysis of large scale three dimensional configurations is usually performed with a time stepping approach based on the adaptation of existing solvers for the unsteady incompressible navier stokes equations we propose instead to solve the nonlinear steady equations with the newton method and to determine the largest growth rate eigenmodes of the linearized equations using a shift and invert spectral transformation and a krylov schur algorithm the solution of the shifted linearized navier stokes problem which is the bottleneck of this approach is computed via an iterative krylov subspace solver preconditioned by the modified augmented lagrangian mal preconditioner benzi et al 2011 the well known efficiency of this preconditioned iterative strategy for solving the real linearized steady state equations is assessed here for the complex shifted linearized equations the effect of various numerical and physical parameters is investigated numerically on a two dimensional flow configuration confirming the reduced number of iterations over state of the art steady state and time stepping based preconditioners a parallel implementation of the steady navier stokes and eigenvalue solvers developed in the freefem language suitably interfaced with the petsc slepc libraries is described and made openly available to tackle three dimensional flow configurations its application on a small scale three dimensional problem shows the good performance of this iterative approach over a direct l u factorization strategy in regards of memory and computational time on a large scale three dimensional problem with 75 million unknowns a 80 parallel efficiency on 256 up to 2048 processes is obtained

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

on element by element schur complement approximations abstract we discuss a methodology to construct sparse approximations of schur complements of two by two block matrices arising in finite element discretizations of partial differential equations earlier results from 2 are extended to more general symmetric positive definite matrices of two by two block form the applicability of the method for general symmetric and nonsymmetric matrices is analysed the paper demonstrates the applicability of the presented method providing extensive numerical experiments

time step restrictions using semi explicit methods for the incompressible navier stokes equations time step restrictions using semi explicit methods for the incompressible navier stokes equations

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