block partitioned solvers for coupled poromechanics a unified framework abstract coupled poromechanical problems appear in a variety of disciplines from reservoir engineering to biomedical applications this work focuses on efficient strategies for solving the matrix systems that result from discretization and linearization of the governing equations these systems have an inherent block structure due to the coupled nature of the mass and momentum balance equations recently several iterative solution schemes have been proposed that exhibit stable and rapid convergence to the coupled solution these schemes appear distinct but a unifying feature is that they exploit the block partitioned nature of the problem to accelerate convergence this paper analyzes several of these schemes and highlights the fundamental connections that underlie their effectiveness we begin by focusing on two specific methods a fully implicit and a sequential implicit scheme in the first approach the system matrix is treated monolithically and a krylov iteration is used to update pressure and displacement unknowns simultaneously to accelerate convergence a preconditioning operator is introduced based on an approximate block factorization of the linear system next we analyze a sequential implicit scheme based on the fixed stress split in this method one iterates back and forth between updating displacement and pressure unknowns separately until convergence to the coupled solution is reached we re interpret this scheme as a block preconditioned richardson iteration and we show that the preconditioning operator is identical to that used within the fully implicit approach rapid convergence in both the richardson and krylov based methods results from a particular choice for a sparse schur complement approximation this analysis leads to a unified framework for developing solution schemes based on approximate block factorizations many classic fully implicit and sequential implicit schemes are simple sub cases the analysis also highlights several new approaches that have not been previously explored for illustration we directly compare the performance and robustness of several variants on a benchmark problem

block diagonal and constraint preconditioners for nonsymmetric indefinite linear systems part i theory we study block diagonal preconditioners and an efficient variant of constraint preconditioners for general two by two block linear systems with zero 2 2 block we derive block diagonal preconditioners from a splitting of the 1 1 block of the matrix from the resulting preconditioned system we derive a smaller so called related system that yields the solution of the original problem solving the related system corresponds to an efficient implementation of constraint preconditioning we analyze the properties of both classes of preconditioned matrices in particular their spectra using analytical results we show that the related system matrix has the more favorable spectrum which in many applications translates into faster convergence for krylov subspace methods we show that fast convergence depends mainly on the quality of the splitting a topic for which a substantial body of theory exists our analysis also provides a number of new relations between block diagonal preconditioners and constraint preconditioners for constrained problems solving the related system produces iterates that satisfy the constraints exactly just as for systems with a constraint preconditioner finally for the lagrange multiplier formulation of a constrained optimization problem we show how scaling nonlinear constraints can dramatically improve the convergence for linear systems in a newton iteration our theoretical results are confirmed by numerical experiments on a constrained optimization problem we consider the general nonsymmetric nonsingular case our only additional requirement is the nonsingularity of the schur complement type matrix derived from the splitting that defines the preconditioners in particular the 1 2 block need not equal the transposed 2 1 block and the 1 1 block might be indefinite or even singular this is the first paper in a two part sequence in the second paper we will study the use of our preconditioners in a variety of applications

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

updating constraint preconditioners for kkt systems in quadratic programming via low rank corrections this work focuses on the iterative solution of sequences of kkt linear systems arising in interior point methods applied to large convex quadratic programming problems this task is the computational core of the interior point procedure and an efficient preconditioning strategy is crucial for the efficiency of the overall method constraint preconditioners are very effective in this context nevertheless their computation may be very expensive for large scale problems and resorting to approximations of them may be convenient here we propose a procedure for building inexact constraint preconditioners by updating a seed constraint preconditioner computed for a kkt matrix at a previous interior point iteration these updates are obtained through low rank corrections of the schur complement of the 1 1 block of the seed preconditioner the updated preconditioners are analyzed both theoretically and computationally the results obtained show that our updating procedure coupled with an adaptive strategy f

efficient solvers for hybridized three field mixed finite element coupled poromechanics abstract we consider a mixed hybrid finite element formulation for coupled poromechanics a stabilization strategy based on a macro element approach is advanced to eliminate the spurious pressure modes appearing in undrained incompressible conditions the efficient solution of the stabilized mixed hybrid block system is addressed by developing a class of block triangular preconditioners based on a schur complement approximation strategy robustness computational efficiency and scalability of the proposed approach are theoretically discussed and tested using challenging benchmark problems on massively parallel architectures

partitioned versus global krylov subspace iterative methods for fe solution of 3 d biot s problem finite element analysis of 3 d biot s consolidation problem needs fast solution of discretized large 2 2 block symmetric indefinite linear systems in this paper partitioned iterative methods and global krylov subspace iterative methods are investigated and compared the partitioned iterative methods considered include stationary partitioned iteration and non stationary prevost s pcg procedure the global krylov subspace methods considered include minres and symmetric qmr sqmr two efficient preconditioners are proposed for global methods numerical experiments based on a pile group problem and simple footing problems with varied soil profiles are carried out numerical results show that when used in conjunction with suitable preconditioners global krylov subspace iterative methods are more promising for large scale computations and further improvement could be possible if significant differences in the solid material properties are addressed in these preconditioned iterative methods 2007 elsevier b v all rights reserved

a relaxed deteriorated pss preconditioner for nonsymmetric saddle point problems from the steady navier stokes equation for nonsymmetric saddle point problems arising from the steady navier stokes equations pan ng and bai presented a deteriorated positive definite and skew hermitian splitting dpss preconditioner pan et al 2006 to accelerate the convergence rates of the krylov subspace iteration methods such as gmres in this paper the unconditional convergence property of the dpss iteration method is proved and a relaxed dpss rdpss preconditioner is proposed the rdpss preconditioner is much closer to the coefficient matrix than the dpss preconditioner in certain norm which straightforwardly results in an rdpss iteration method the convergence conditions of the rdpss iteration are analyzed and the optimal parameter which minimizes the spectral radius of the rdpss iteration matrix is derived using the rdpss preconditioner to accelerate some krylov subspace methods like gmres is also studied the eigenproperty of the preconditioned matrix is described and an upper bound of the degree of the minimal polynomial of the preconditioned matrix is obtained finally numerical experiments of a model navier stokes equation are presented to illustrate the efficiency of the rdpss preconditioner

a multiscale fixed stress split iterative scheme for coupled flow and poromechanics in deep subsurface reservoirs abstract in coupled flow and poromechanics phenomena representing hydrocarbon production or co 2 sequestration in deep subsurface reservoirs the spatial domain in which fluid flow occurs is usually much smaller than the spatial domain over which significant deformation occurs the typical approach is to either impose an overburden pressure directly on the reservoir thus treating it as a coupled problem domain or to model flow on a huge domain with zero permeability cells to mimic the no flow boundary condition on the interface of the reservoir and the surrounding rock the former approach precludes a study of land subsidence or uplift and further does not mimic the true effect of the overburden on stress sensitive reservoirs whereas the latter approach has huge computational costs in order to address these challenges we augment the fixed stress split iterative scheme with upscaling and downscaling operators to enable modeling flow and mechanics on overlapping nonmatching hexahedral grids flow is solved on a finer mesh using a multipoint flux mixed finite element method and mechanics is solved on a coarse mesh using a conforming galerkin method the multiscale operators are constructed using a procedure that involves singular value decompositions a surface intersections algorithm and delaunay triangulations we numerically demonstrate the convergence of the augmented scheme using the classical mandel s problem solution

parameterized approximate block lu preconditioners for generalized saddle point problems abstract in this paper we are concerned with the iteration solution of generalized saddle point problems based on the exact block lu factorization of the coefficient matrix we construct a class of parameterized approximate block lu factorization preconditioners which rely on suitable approximations of the schur complement of the 1 1 block of the coefficient matrix convergence of the corresponding iteration methods is analyzed and the optimal iteration parameters minimizing the spectral radii are deduced algebraic characteristics of the related preconditioned matrices are discussed including eigenvalue and eigenvector distributions and upper bounds for degree of the minimal polynomial the established results extend those of the approximate factorization and variants of the hermitian and skew hermitian splitting and positive and skew hermitian splitting preconditioners for saddle point problems numerical experiments are demonstrated to illustrate the efficiency of the new preconditioners

eigenvalue estimates of an indefinite block triangular preconditioner for saddle point problems in this paper an indefinite block triangular preconditioner for symmetric saddle point problems is discussed the bounds for all the real eigenvalues of the preconditioned matrix are provided and the bounds for the real and imaginary parts of all the complex eigenvalues of the preconditioned matrix are also provided the corresponding theoretical results of paper by bergamaschi 2012 are improved numerical experiments are reported to confirm our theoretical results

non linear elliptic and parabolic equations involving measure data in this paper we prove the existence of solutions for equations of the type div a du f in a bounded open set u 0 on where a is a possibly non linear function satisfying some coerciveness and monotonicity assumptions and f is a bounded measure we also consider the equation div a du g u f in u 0 on with f l1 or f m u u 0 and the parabolic equivalent of the first elliptic equation

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

guaranteed and computable error bounds for approximations constructed by an iterative decoupling of the biot problem abstract the paper is concerned with guaranteed a posteriori error estimates for a class of evolutionary problems related to poroelastic media governed by the quasi static linear biot equations the system is decoupled by employing the fixed stress split scheme which leads to an iteratively solved semi discrete system the error bounds are derived by combining a posteriori estimates for contractive mappings with functional type error control for elliptic partial differential equations the estimates are applicable to any approximation in the admissible functional space and are independent of the discretization method they are fully computable do not contain mesh dependent constants and provide reliable global estimates of the error measured in the energy norm moreover they suggest efficient error indicators for the distribution of local errors and can be used in adaptive procedures

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

preconditioned hermitian and skew hermitian splitting methods for non hermitian positive semidefinite linear systems for the positive semidefinite system of linear equations of a block two by two structure by making use of the hermitian skew hermitian splitting iteration technique we establish a class of preconditioned hermitian skew hermitian splitting iteration methods theoretical analysis shows that the new method converges unconditionally to the unique solution of the linear system moreover the optimal choice of the involved iteration parameter and the corresponding asymptotic convergence rate are computed exactly numerical examples further confirm the correctness of the theory and the effectiveness of the method

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

two way coupling in reservoir geomechanical models vertex centered galerkin geomechanical model cell centered and vertex centered finite volume reservoir models procedures to couple reservoir and geomechanical models are reviewed the focus is on immiscible compressible non compositional reservoir geomechanical models such models require the solution to coupled stress pressure saturation and temperature equations although the couplings between saturation and temperature with stress and fluid pressure are weak and can be adequately captured thru staggered fixed point iterations the couplings between stress and pressure equations are strong and require special procedures for accurate integration as shown and discussed in detail in prevost 2013 two way coupling i e simultaneous integration of pressure and stress equations is required if poromechanical effects are to be captured accurately in our previous work a galerkin implementation of both pressure and stress equations was used with equal order interpolant for both pressure and solid displacements

convergence analysis of two grid fixed stress split iterative scheme for coupled flow and deformation in heterogeneous poroelastic media abstract we perform a convergence analysis of a two grid fixed stress split algorithm for the biot system modeling coupled flow and deformation in heterogeneous poroelastic media the two grid fixed stress split scheme solves the flow subproblem on a fine grid using a multipoint flux mixed finite element method by imposing the fixed mean stress constraint followed by the poromechanics subproblem on a coarse grid using a conforming galerkin method in every coupling iteration at each time step restriction operators map the fine scale flow solution to the coarse scale poromechanical grid and prolongation operators map the coarse scale poromechanical solution to the fine scale flow grid the coupling iterations are repeated until convergence and backward euler is employed for time marching the convergence analysis is based on studying the equations satisfied by the difference of iterates to show that the two grid scheme is a contraction map under certain conditions those conditions are used to construct the restriction and prolongation operators as well as arrive at effective elastic properties for the coarse scale poromechanical solve in terms of the fine scale elastic properties we analyze the contraction map with a numerical result comparing the numerically computed to the theoretically obtained contraction constant for a poroelastic medium with substantial spatial variability in the poroelastic moduli

an inexact primal dual path following algorithm for convex quadratic sdp we propose primal dual path following mehrotra type predictor corrector methods for solving convex quadratic semidefinite programming qsdp problems of the form min x frac 1 2 x bullet mathcal q x c bullet x mathcal a x b x succeq 0 where mathcal q is a self adjoint positive semidefinite linear operator on mathcal s n b r m and mathcal a is a linear map from mathcal s n to r m at each interior point iteration the search direction is computed from a dense symmetric indefinite linear system called the augmented equation of dimension m n n 1 2 such linear systems are typically very large and can only be solved by iterative methods we propose three classes of preconditioners for the augmented equation and show that the corresponding preconditioned matrices have favorable asymptotic eigenvalue distributions for fast convergence under suitable nondegeneracy assumptions numerical experiments on a variety of qsdps with n up to 1600 are performed and the computational results show that our methods are efficient and robust

paradigms and puzzles in the theory of dynamical systems a self contained exposition is given of an approach to mathematical models in particular to the theory of dynamical systems the basic ingredients form a triptych with the behavior of a system in the center and behavioral equations with latent variables as side panels the author discusses a variety of representation and parametrization problems in particular questions related to input output and state models the proposed concept of a dynamical system leads to a new view of the notions of controllability and observability and of the interconnection of systems in particular to what constitutes a feedback control law the final issue addressed is that of system identification it is argued that exact system identification leads to the question of computing the most powerful unfalsified model

a behavioral approach to control of distributed systems this paper develops a theory of control for distributed systems i e those defined by systems of constant coefficient partial differential operators via the behavioral approach of willems the study here is algebraic in the sense that it relates behaviors of distributed systems to submodules of free modules over the polynomial ring in several indeterminates as in the lumped case behaviors of distributed arma systems can be reduced to ar behaviors this paper first studies the notion of ar controllable distributed systems following the corresponding definition for lumped systems due to willems it shows that as in the lumped case the class of controllable ar systems is precisely the class of ma systems it then shows that controllable 2 d distributed systems are necessarily given by free submodules whereas this is not the case for n d distributed systems n ge 3 this therefore points out an important difference between these two cases this paper then defines two notions of autonomous distributed systems which mimic different properties of lumped autonomous systems control is the process of restricting a behavior to a specific desirable autonomous subbehavior a notion of stability generalizing bounded input bounded output stability of lumped systems is proposed and the pole placement problem is defined for distributed systems this paper then solves this problem for a class of distributed behaviors

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

finite volume hydromechanical simulation in porous media cell centered finite volume methods are prevailing in numerical simulation of flow in porous media however due to the lack of cell centered finite volume methods for mechanics coupled flow and deformation is usually treated either by coupled finite volume finite element discretizations or within a finite element setting the former approach is unfavorable as it introduces two separate grid structures while the latter approach loses the advantages of finite volume methods for the flow equation recently we proposed a cell centered finite volume method for elasticity herein we explore the applicability of this novel method to provide a compatible finite volume discretization for coupled hydromechanic flows in porous media we detail in particular the issue of coupling terms and show how this is naturally handled furthermore we observe how the cell centered finite volume framework naturally allows for modeling fractured and fracturing porous media through internal boundary conditions we support the discussion with a set of numerical examples the convergence properties of the coupled scheme are first investigated second we illustrate the practical applicability of the method both for fractured and heterogeneous media

stabilized mixed continuous enriched galerkin formulations for locally mass conservative poromechanics abstract local element wise mass conservation is often highly desired for numerical solutions to coupled poromechanical problems as an efficient numerical method featuring this property mixed continuous galerkin cg enriched galerkin eg finite elements have recently been proposed in which piecewise constant functions are enriched to the pore pressure interpolation functions of the conventional mixed cg cg elements while this enrichment of the pressure space provides local mass conservation it unavoidably alters the stability condition for mixed finite elements because no stabilization method has been available for the new stability condition high order displacement interpolation has been required for mixed cg eg elements if undrained condition is expected to circumvent this requirement here we develop stabilized formulations for the mixed cg eg elements that permit equal order interpolation functions even in the undrained limit we begin by identifying the inf sup condition for mixed cg eg elements by phrasing an enriched poromechanical problem as a twofold saddle point problem we then derive two types of stabilized formulations one based on the polynomial pressure projection ppp method and the other based on the fluid pressure laplacian fpl method a key finding of this work is that both methods lead to stabilization terms that should be augmented only to the cg part of the pore pressure field not to the enrichment part the two stabilized formulations are verified and investigated through numerical examples that involve various conditions spanning 1d to 3d isotropy to anisotropy and homogeneous to heterogeneous domains the methodology presented in this work may also help stabilize other types of mixed finite elements in which the constraint field is enriched by additional functions

block preconditioners for symmetric indefinite linear systems this paper presents a systematic theoretical and numerical evaluation of three common block preconditioners in a krylov subspace method for solving symmetric indefinite linear systems the focus is on large scale real world problems where block approximations are a practical necessity the main illustration is the performance of the block diagonal constrained and lower triangular preconditioners over a range of block approximations for the symmetric indefinite system arising from large scale finite element discretization of biot s consolidation equations this system of equations is of fundamental importance to geomechanics numerical studies show that simple diagonal approximations to the 1 1 block k and inexpensive approximations to the schur complement matrix s may not always produce the most spectacular time savings when k is explicitly available but is able to deliver reasonably good results on a consistent basis in addition the block diagonal preconditioner with a negative 2 2 block appears to be reasonably competitive when compared to the more complicated ones these observation are expected to remain valid for coefficient matrices whereby the 1 1 block is sparse diagonally significant a notion weaker than diagonal dominance moderately well conditioned and has a much larger block size than the 2 2 block copyright 2004 john wiley sons ltd

a note on the positive stable block triangular preconditioner for generalized saddle point problems abstract in this paper we give a sharp eigenvalue bound for the positive stable block triangular preconditioned matrix presented in a recent paper by cao z h cao positive stable block triangular preconditioners for symmetric saddle point problems appl numer math 57 2007 899 910 the intervals containing these eigenvalues of the preconditioned matrix remove the origin which is benefit for further study numerical experiments of a model stokes problem are presented to show the estimate and the effectiveness of the positive stable block triangular preconditioners

a modified ssor preconditioner for sparse symmetric indefinite linear systems of equations the standard ssor preconditioner is ineffective for the iterative solution of the symmetric indefinite linear systems arising from finite element discretization of the biot s consolidation equations in this paper a modified block ssor preconditioner combined with the eisenstat trick implementation is proposed for actual implementation a pointwise variant of this modified block ssor preconditioner is highly recommended to obtain a compromise between simplicity and effectiveness numerical experiments show that the proposed modified ssor preconditioned symmetric qmr solver can achieve faster convergence than several effective preconditioners published in the recent literature in terms of total runtime moreover the proposed modified ssor preconditioners can be generalized to non symmetric biot s systems copyright 2005 john wiley sons ltd

on the sets of l regularity of solutions for a class of degenerate nonlinear problems with slightly regular data we consider the dirichlet problem for a class of degenerate nonlinear elliptic fourth order equations in divergence form coefficients of the equations satisfy a strengthened ellipticity condition involving two weighted functions the right hand sides f x u of the equations depend on the unknown function u under some conditions including lm regularity of f 0 with m close to 1 we establish that the problem under consideration has a w solution bounded on the sets where the behaviour of the data of the problem is regular enough

optimal accelerated sor like asor method for singular symmetric saddle point problems abstract in a recent paper a new iterative method for the solution of the nonsingular symmetric saddle point problem was proposed njeru and guo 2016 the asor method belongs to the family of the sor like methods and uses two parameters and convergence intervals for the parameters involved were found in the present work we analyze and study an extension of the above problem to the singular case and determine optimal values for the two parameters as well as for the semi convergence factor of the asor method numerical results are presented to show the efficiency of the optimal singular asor method

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

stability of multirate explicit coupling of geomechanics with flow in a poroelastic medium abstract we consider single rate and multirate explicit schemes for the biot system modeling coupled flow and geomechanics in a poro elastic medium these schemes are widely used in practice that follows a sequential procedure in which the flow and mechanics problems are fully decoupled in such a scheme the flow problem is solved first with time lagging the displacement term followed by the mechanics solve the multirate explicit coupling scheme exploits the different time scales for the mechanics and flow problems by taking multiple finer time steps for flow within one coarse mechanics time step we provide fully discrete schemes for both the single and multirate approaches that use backward euler time discretization and mixed spaces for flow and conformal galerkin for mechanics we perform a rigorous stability analysis and derive the conditions on reservoir parameters and the number of finer flow solves to ensure stability for both schemes furthermore we investigate the computational time savings for explicit coupling schemes against iterative coupling schemes

unconditionally stable sequential schemes for all way coupled thermoporomechanics undrained adiabatic and extended fixed stress splits abstract we propose two sequential methods for the all way coupled thermoporomechanical problems the undrained adiabatic split that combines together the undrained and adiabatic methods of poromechanics and thermomechanics respectively and the extended fixed stress split we perform nonlinear stability analysis for the two sequential methods introducing a new extended norm that can encompass the all way coupled thermoporomechanical problems we find that the two sequential methods are unconditionally stable if 0 5 where is the parameter of time discretization for the generalized midpoint rule at t n we also perform the spectral analysis in order to further investigate stability and convergence of the two methods the spectral analysis shows unconditional stability of the two sequential methods which yields the magnitudes of the amplification factor less than unity we then show numerical examples which support the a priori stability and convergence estimates

on augmentation block triangular preconditioners for regularized saddle point problems two augmentation block triangular preconditioners were introduced by shen et al 2012 for the regularized saddle point problem however the spectral analysis of the preconditioner based on the augmentation of the 2 2 block was not throughly derived there in this short paper we give a detailed spectral analysis on this preconditioner a numerical example is employed to validate the efficiency of the presented theoretical results

novel preconditioners for the iterative solution to fe discretized coupled consolidation equations a major computational issue in the finite element fe integration of coupled consolidation equations is the repeated solution in time of the resulting discretized indefinite system because of ill conditioning the iterative solution which is recommended in large size 3d settings requires the computation of a suitable preconditioner to guarantee convergence in this paper the coupled system is solved by a krylov subspace method preconditioned by an inexact constraint preconditioner icp preserving the same block structure as the native fe matrix the conditioning number of the preconditioned coupled problem depends on the quality of the approximation of the block corresponding to the structural stiffness matrix an efficient algorithm to implement icp into a krylov subspace method is developed numerical tests performed on realistic 3d problems reveal that icp typically outperforms standard ilut preconditioners and proves much more robust in severely ill conditioned problems

performance and robustness of block constraint preconditioners in finite element coupled consolidation problems block constraint preconditioners are a most recent development for the iterative solution to large scale often ill conditioned coupled consolidation problems a major limitation to their practical use however is the somewhat difficult selection of a number of user defined parameters at least 4 in a more or less optimal way the present paper investigates the robustness of three variants of the block constraint preconditioning in relation to the above parameters a theoretical analysis of the eigenspectrum of the preconditioned matrix provides relatively simple bounds of the eigenvalues as a function of these parameters a number of test problems used to validate the theoretical results show that both the mixed constraint preconditioner mcp combined with the symmetric quasi minimal residual sqmr solver and the mcp triangular variant t mcp combined with the bi conjugate gradient stabilized bi cgstab are efficient and robust tools for the solution to difficult finite element discretized coupled consolidation problems moreover the practical selection of the user defined parameters is relatively easy as a stable behavior is observed for a wide range of fill in degree values the theoretical bounds on the eigenspectrum of the preconditioned matrix may help to suggest the most appropriate parameter combination copyright 2009 john wiley sons ltd

mixed constraint preconditioning in computational contact mechanics contact mechanics can be addressed numerically by finite elements using either a penalty formulation or the lagrange multipliers the penalty approach leads to a linearized symmetric positive definite system which can prove severely ill conditioned with the iterative solution to large 3d problems requiring expensive preconditioners to accelerate or even to allow for convergence in the present paper a mixed constraint preconditioner mcp is developed and implemented for the iterative solution of contact problems based on the penalty approach a theoretical bound for the eigenspectrum of the preconditioned matrix is provided depending on the user specified parameters that control the preconditioner fill in the performance of mcp combined with the preconditioned conjugate gradient is studied in two real 3d test cases describing the geomechanics of faulted porous media and compared to a well established preconditioner such as the incomplete cholesky decomposition illt with optimal fill in numerical results show that typically mcp outperforms illt especially in the most challenging problems where very large penalty parameters must be used finally a modification aimed at simplifying the mcp implementation and increasing its robustness is developed and experimented with showing that the modified mcp can be viewed as an efficient and viable alternative to both mcp and illt especially in problems where the size of the contact surfaces is large with respect to the global domain

on solving block structured indefinite linear systems we consider 2 2 block indefinite linear systems whose 2 2 block is zero such systems arise in many applications we discuss two techniques that are based on modifying the 1 1 block in a way that makes the system easier to solve the main part of the paper focuses on an augmented lagrangian approach a technique that modifies the 1 1 block without changing the system size the choice of the parameter involved the spectrum of the linear system and its condition number are discussed and some analytical observations are provided a technique of deflating the 1 1 block is then introduced finally numerical experiments that validate the analysis are presented

monolithic and splitting solution schemes for fully coupled quasi static thermo poroelasticity with nonlinear convective transport abstract this paper concerns monolithic and splitting based iterative procedures for the coupled nonlinear thermo poroelasticity model problem the thermo poroelastic model problem we consider is formulated as a three field system of pde s consisting of an energy balance equation a mass balance equation and a momentum balance equation where the primary variables are temperature fluid pressure and elastic displacement due to the presence of a nonlinear convective transport term in the energy balance equation it is convenient to have access to both the pressure and temperature gradients hence we introduce these as two additional variables and extend the original three field model to a five field model for the numerical solution of this five field formulation we compare six approaches that differ by how we treat the coupling decoupling between the flow and from heat and from the mechanics suitable for varying coupling strength between the three physical processes the approaches have in common a simultaneous application of the so called l scheme which works both to stabilize iterative splitting as well as to linearize nonlinear problems and can be seen as a generalization of the undrained and fixed stress split algorithms more precisely the derived procedures transform a nonlinear and fully coupled problem into a set of simpler subproblems to be solved sequentially in an iterative fashion we provide a convergence proof for the derived algorithms and demonstrate their performance through several numerical examples investigating different strengths of the coupling between the different processes

space time finite element approximation of the biot poroelasticity system with iterative coupling abstract we analyze an optimized artificial fixed stress iterative scheme for a space time finite element approximation of the biot system modeling fluid flow in deformable porous media the iteration is based on a prescribed constant artificial volumetric mean total stress in the first half step the optimization comes through the adaptation of a numerical stabilization or tuning parameter and aims at an acceleration of the iterations the separated subproblems of fluid flow written as a mixed first order in space system and mechanical deformation are discretized by space time finite element methods of arbitrary order continuous and discontinuous galerkin discretizations of the time variable are encountered the convergence of the iterative schemes is proved for the continuous and fully discrete case the choice of the optimization parameter is identified in the proofs of convergence of the iterations the analyses are illustrated and confirmed by numerical experiments

on generalized successive overrelaxation methods for augmented linear systems for the augmented system of linear equations golub wu and yuan recently studied an sor like method bit 41 2001 71 85 by further accelerating it with another parameter in this paper we present a generalized sor gsor method for the augmented linear system we prove its convergence under suitable restrictions on the iteration parameters and determine its optimal iteration parameters and the corresponding optimal convergence factor theoretical analyses show that the gsor method has faster asymptotic convergence rate than the sor like method also numerical results show that the gsor method is more effective than the sor like method when they are applied to solve the augmented linear system this gsor method is further generalized to obtain a framework of the relaxed splitting iterative methods for solving both symmetric and nonsymmetric augmented linear systems by using the techniques of vector extrapolation matrix relaxation and inexact iteration besides we also demonstrate a complete version about the convergence theory of the sor like method

an algebraic multigrid method for linear elasticity we present an algebraic multigrid amg method for the efficient solution of linear block systems stemming from a discretization of a system of partial differential equations pdes it generalizes the classical amg approach for scalar problems to systems of pdes in a natural blockwise fashion we apply this approach to linear elasticity and show that the block interpolation described in this paper reproduces the rigid body modes i e the kernel elements of the discrete linear elasticity operator it is well known from geometric multigrid methods that this reproduction of the kernel elements is an essential property to obtain convergence rates which are independent of the problem size we furthermore present results of various numerical experiments in two and three dimensions they confirm that the method is robust with respect to variations of the poisson ratio nu we obtain rates rho 0 4 for nu 0 4 these measured rates clearly show that the method provides fast convergence for a large variety of discretized elasticity problems

an efficient diagonal preconditioner for finite element solution of biot s consolidation equations finite element simulations of very large scale soil structure interaction problems e g excavations tunnelling pile rafts etc typically involve the solution of a very large ill conditioned and indefinite biot system of equations the traditional preconditioned conjugate gradient solver coupled with the standard jacobi sj preconditioner can be very inefficient for this class of problems this paper presents a robust generalized jacobi gj preconditioner that is extremely effective for solving very large scale biot s finite element equations using the symmetric quasi minimal residual method the gj preconditioner can be formed inverted and implemented within an element by element framework as readily as the sj preconditioner it was derived as a diagonal approximation to a theoretical form which can be proven mathematically to possess an attractive eigenvalue clustering property the effectiveness of the gj preconditioner over a wide range of soil stiffness and permeability was demonstrated numerically using a simple three dimensional footing problem this paper casts a new perspective on the potentialities of the simple diagonal preconditioner which has been commonly perceived as being useful only in situations where it can serve as an approximate inverse to a diagonally dominant coefficient matrix copyright 2002 john wiley sons ltd

stability and convergence of sequential methods for coupled flow and geomechanics fixed stress and fixed strain splits abstract we analyze stability and convergence of sequential implicit methods for coupled flow and geomechanics in which the flow problem is solved first we employ the von neumann and energy methods for linear and nonlinear problems respectively we consider two sequential methods with the generalized midpoint rule for tn where is the parameter of time discretization namely the fixed strain and fixed stress splits the von neumann method indicates that the fixed strain split is only conditionally stable and that its stability limit is a coupling strength less than unity if 0 5 on the other hand the fixed stress split is unconditionally stable when 0 5 the amplification factors of the fixed stress split are different from those of the undrained split and are identical to the fully coupled method unconditional stability of the fixed stress split is also obtained from the energy method for poroelastoplasticity we show that the fixed stress split is contractive and b stable when 0 5 we also estimate the convergence behaviors for the two sequential methods by the matrix based and spectral analyses for the backward euler method in time from the estimates the fixed strain split may not be convergent with a fixed number of iterations particularly around the stability limit even though it is stable the fixed stress split however is convergent for a fixed number of iterations showing better accuracy than the undrained split even when we cannot obtain the exact local bulk modulus or exact rock compressibility at the flow step a priori due to complex boundary conditions or the nonlinearity of the materials the fixed stress split can still provide stability and convergence by an appropriate estimation of the local bulk modulus such as the dimension based estimation by which the employed local bulk modulus is less stiff than the exact local bulk modulus we provide numerical examples supporting all the estimates of stability and convergence for the fixed strain and fixed stress splits

a new partitioned staggered scheme for flexible multibody interactions with strong inertial effects abstract in generic flexible multibody interaction problems the differences in the mass damping and stiffness of the bodies can be very large of particular interest is a small change in the higher mass structure which triggers relatively large inertial effects on the lower mass structure in this work a new partitioned staggered time integration scheme is developed and its applicability is extended to the problems involving low mass ratios in coupled multibody problems the equation of motion of body with larger inertia is integrated by implicit newmark method whereas the smaller inertia is solved using a robust self starting explicit integration procedure the coupled formulation includes explicit correction terms that adjusts the amount of interfacial velocity which plays a key role in the stability and accuracy of the simulations a wide range of mass 10 6 10 1 damping and stiffness 10 3 10 3 ratios are chosen to assess the stability and accuracy characteristics of the proposed scheme the closed form expressions for the coupling parameter have been constructed for both matching and non matching time stepping through the godunov ryabenkii normal mode analysis the stability of the proposed method is observed as a function of the relative properties and temporal discretisation of the coupled system the time step of the heavier body significantly influences the stability more than the lighter body to maintain the staggering error minimal an optimal range of the coupling parameter is identified the error computed with uniform variation in the time step revealed that the present method is more accurate than the existing methods the partitioned method is an energy preserving integration method for the dynamic analysis of multibody systems as a potential application of this scheme flexible multibody problems in the field of offshore engineering viz wave energy converter and floater mooring systems are presented and validated with experimental measurements

mixed constraint preconditioners for the iterative solution of fe coupled consolidation equations the finite element fe integration of the coupled consolidation equations requires the solution of linear symmetric systems with an indefinite saddle point coefficient matrix because of ill conditioning the repeated solution in time of the fe equations may be a major computational issue requiring ad hoc preconditioning strategies to guarantee the efficient convergence of krylov subspace methods in the present paper a mixed constraint preconditioner mcp is developed combining implicit and explicit approximations of the inverse of the structural sub matrix with the performance investigated in some representative examples an upper bound of the eigenvalue distance from unity is theoretically provided in order to give practical indications on how to improve the preconditioner the mcp is efficiently implemented into a krylov subspace method with the performance obtained in 2d and 3d examples compared to that of inexact constraint preconditioners and least square logarithm scaled ilut preconditioners two variants of mcp t mcp and d mcp developed with the aim at reducing the cost of the preconditioner application are also tested the results show that the mcp variants constitute a reliable and robust approach for the efficient solution of realistic coupled consolidation fe models and especially so in severely ill conditioned problems

minres qlp a krylov subspace method for indefinite or singular symmetric systems cg symmlq and minres are krylov subspace methods for solving symmetric systems of linear equations when these methods are applied to an incompatible system that is a singular symmetric least squares problem cg could break down and symmlq s solution could explode while minres would give a least squares solution but not necessarily the minimum length pseudoinverse solution this understanding motivates us to design a minres like algorithm to compute minimum length solutions to singular symmetric systems minres uses qr factors of the tridiagonal matrix from the lanczos process where r is upper tridiagonal minres qlp uses a qlp decomposition where rotations on the right reduce r to lower tridiagonal form on ill conditioned systems singular or not minres qlp can give more accurate solutions than minres we derive preconditioned minres qlp new stopping rules and better estimates of the solution and residual norms the matrix norm and the condition number

stability and convergence of sequential methods for coupled flow and geomechanics drained and undrained splits abstract we perform a stability and convergence analysis of sequential methods for coupled flow and geomechanics in which the mechanics sub problem is solved first we consider slow deformations so that inertia is negligible and the mechanical problem is governed by an elliptic equation we use biot s self consistent theory to obtain the classical parabolic type flow problem we use a generalized midpoint rule parameter between 0 and 1 time discretization and consider two classical sequential methods the drained and undrained splits the von neumann method provides sharp stability estimates for the linear poroelasticity problem the drained split with backward euler time discretization 1 is conditionally stable and its stability depends only on the coupling strength and it is independent of time step size the drained split with the midpoint rule 0 5 is unconditionally unstable the mixed time discretization with 1 0 for mechanics and 0 5 for flow has the same stability properties as the backward euler scheme the von neumann method indicates that the undrained split is unconditionally stable when 0 5 we extend the stability analysis to the nonlinear regime poro elastoplasticity via the energy method it is well known that the drained split does not inherit the contractivity property of the continuum problem thereby precluding unconditional stability for the undrained split we show that it is b stable therefore unconditionally stable at the algorithmic level when 0 5 we also analyze convergence of the drained and undrained splits and derive the a priori error estimates from matrix algebra and spectral analysis we show that the drained split with a fixed number of iterations is not convergent even when it is stable the undrained split with a fixed number of iterations is convergent for a compressible system i e finite biot modulus for a nearly incompressible system i e very large biot modulus the undrained split loses first order accuracy and becomes non convergent in time we also study the rate of convergence of both splits when they are used in a fully iterated sequential scheme when the medium permeability is high or the time step size is large which corresponds to a high diffusion of pressure the error amplification of the drained split is lower and therefore converges faster than the undrained split the situation is reversed in the case of low permeability and small time step size we provide numerical experiments supporting all the stability and convergence estimates of the drained and undrained splits in the linear and nonlinear regimes we also show that our spatial discretization finite volumes for flow and finite elements for mechanics removes the well documented spurious instability in consolidation problems at early times

on the fixed stress split scheme as smoother in multigrid methods for coupling flow and geomechanics the fixed stress split method has been widely used as solution method in the coupling of flow and geomechanics in this work we analyze the behavior of an inexact version of this algorithm as smoother within a geometric multigrid method in order to obtain an efficient monolithic solver for biot s problem this solver combines the advantages of being a fully coupled method with the benefit of decoupling the flow and the mechanics part in the smoothing algorithm moreover the fixed stress split smoother is based on the physics of the problem and therefore all parameters involved in the relaxation are based on the physical properties of the medium and are given a priori a local fourier analysis is applied to study the convergence of the multigrid method and to support the good convergence results obtained the proposed geometric multigrid algorithm is used to solve several tests in semi structured triangular grids in order to show the good behavior of the method and its practical utility

a preconditioning technique for indefinite systems resulting from mixed approximations of elliptic problems this paper provides a preconditioned iterative technique for the solution of saddle point problems these problems typically arise in the numerical approximation of partial differential equations by lagrange multiplier techniques and or mixed methods the saddle point problem is reformulated as a symmetric positive definite system which is then solved by conjugate gradient iteration applications to the equations of elasticity and stokes are discussed and the results of numerical experiments are given

on parameterized block triangular preconditioners for generalized saddle point problems in this paper we discuss two classes of parameterized block triangular preconditioners for the generalized saddle point problems these preconditioners generalize the common block diagonal and triangular preconditioners we will give distributions of the eigenvalues of the preconditioned matrix and provide estimates for the interval containing the real eigenvalues numerical experiments of a model stokes problem are presented

parameter estimates for the relaxed dimensional factorization preconditioner and application to hemodynamics we present new results on the relaxed dimensional factorization rdf preconditioner for solving saddle point problems from incompressible flow simulations first introduced in benzi et al 2011 this method contains a parameter 0 0 to be chosen by the user previous works provided an estimate of in the 2d case using local fourier analysis a novel algebraic estimation technique for finding a suitable value of the rdf parameter in both the 2d and the 3d case with arbitrary geometries is proposed this technique is tested on a variety of discrete saddle point problems arising from the approximation of the navier stokes equations using a marker and cell scheme and a finite element one we also show results for a large scale problem relevant for hemodynamics simulation that we solve in parallel using up to 8196 cores

two new variants of the hss preconditioner for regularized saddle point problems two new preconditioners which can be viewed as improved variants of the hermitian and skew hermitian splitting hss preconditioner are presented for regularized saddle point problems the unconditionally convergent properties of the corresponding iteration methods are deduced and the selection of optimal iteration parameters resulting in fast convergence for the two iteration methods are discussed moreover eigenvalue bounds of the preconditioned matrices and upper bounds of the degree of their minimal polynomials are obtained compared with the hss preconditioner they are much better approximations to the coefficient matrix of the saddle point problem and they have better convergence properties and spectrum distributions numerical experiments from the stokes problem are presented which illustrates the advantages of the two preconditioners over the hss preconditioner and some other preconditioners to accelerate the convergence rate of gmres

spectral properties of a class of matrix splitting preconditioners for saddle point problems based on the accelerated hermitian and skew hermitian splitting iteration scheme bai and golub 2007 we propose a new two parameter matrix splitting preconditioner in this paper spectral properties of the preconditioned matrix are analyzed in detail furthermore based on this preconditioner an improved version of matrix splitting preconditioner is presented and analyzed finally performance of the preconditioners is compared by using gmres m as an iterative solver on linear systems arising from the discretization of stokes and navier stokes equations

taylor approximations of multidimensional linear differential systems abstractthe taylor approximations of a multidimensional linear differential system are of importance as they contain a complete information about it it is shown that in order to construct them it is sufficient to truncate the exponential trajectories only a computation of the taylor approximations is provided using purely algebraic means without requiring explicit knowledge of the trajectories

on the eigenvalues of a class of saddle point matrices we study spectral properties of a class of block 2 2 matrices that arise in the solution of saddle point problems these matrices are obtained by a sign change in the second block equation of the symmetric saddle point linear system we give conditions for having a positive real spectrum and for ensuring diagonalizability of the matrix in particular we show that these properties hold for the discrete stokes operator and we discuss the implications of our characterization for augmented lagrangian formulations for krylov subspace solvers and for certain types of preconditioners

a note on ehrenpreis fundamental principle abstract ehrenpreis fundamental principle is an existence theorem for inhomogeneous linear partial differential equations and has great system theoretic significance usually it is formulated for the equations defined on open convex sets however openness of domains of definition is somewhat restrictive in systems theory in this article we show that the principle is valid in the case of sets that are the unions of increasing sequences of convex compact sets moreover we offer a simplified proof

simple like preconditioners for saddle point problems from the steady navier stokes equations the semi implicit method for pressure linked equations simple preconditioner is further generalized to a class of simple like sl preconditioners for solving saddle point problems from the steady navier stokes equations the sl preconditioners can be also viewed as a generalization of the relaxed deteriorated pss rdpss preconditioner proposed by cao et al 2015 convergence analysis of the corresponding sl iteration is presented and the optimal iteration parameter is obtained by minimizing the spectral radius of the sl iteration matrix moreover krylov subspace acceleration of the sl preconditioning is studied the sl preconditioned saddle point matrix is analyzed results about eigenvalue and eigenvector distributions and the minimal polynomial are derived numerical experiments from leaky two dimensional lid driven cavity problems are given which show the advantages of the sl preconditioned gmres over the dpss and rdpss preconditioned ones for solving saddle point problems