implicit finite difference approximation for time fractional diffusion equations time fractional diffusion equations are used when attempting to describe transport processes with long memory where the rate of diffusion is inconsistent with the classical brownian motion model in this paper we develop an implicit unconditionally stable numerical method to solve the one dimensional linear time fractional diffusion equation formulated with caputo s fractional derivative on a finite slab several numerical examples of interest are also included

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

multi level incomplete factorizations for the iterative solution of non linear finite element problems several engineering applications give rise quite naturally to linearized fe systems of equations possessing a multi level structure an example is provided by geomechanical models of layered and faulted geological formations for such problems the use of a multi level incomplete factorization mif as a preconditioner for krylov subspace methods can prove a robust and efficient solution accelerator allowing for a fine tuning of the fill in degree with a significant improvement in both the solver performance and the memory consumption the present paper develops two novel mif variants for the solution of multi level symmetric positive definite systems two correction algorithms are proposed with the aim of preserving the positive definiteness of the preconditioner thus avoiding possible breakdowns of the preconditioned conjugate gradient solver the mif variants are experimented with in the solution of both a single system and a long term quasi static simulation dealing with a multi level geomechanical application the numerical results show that mif typically outperforms by up to a factor 3 a more traditional algebraic preconditioner such as an incomplete cholesky factorization with partial fill in the advantage is emphasized in a long term simulation where the fine fill in tuning allowed for by mif yields a significant improvement for the computer memory requirement as well copyright 2009 john wiley sons ltd

existence and uniqueness of travelling waves for a neural network a one dimensional scalar neural network with two stable steady states is analysed it is shown that there exists a unique monotone travelling wave front which joins the two stable states some additional properties of the wave such as the direction of its velocity are discussed

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

interface fluctuations for the d 1 stochastic ginzburg landau equation with nonsymmetric reaction term we consider a ginzburg landau equation in the interval e 1 e 1 e 0 with neumann boundary conditions perturbed by an additive white noise of strength sqrt varepsilon and reaction term being the derivative of a function which has two equal depth wells at 1 but is not symmetric when e 0 the equation has equilibrium solutions that are increasing and connect 1 with 1 we call them instantons and we study the evolution of the solutions of the perturbed equation in the limit e 0 when the initial datum is close to an instanton we prove that for times that may be of the order of e 1 the solution stays close to some instanton whose center suitably normalized converges to a brownian motion plus a drift this drift is known to be zero in the symmetric case and using a perturbative analysis we show that if the nonsymmetric part of the reaction term is sufficiently small it determines the sign of the drift

fsai based parallel mixed constraint preconditioners for saddle point problems arising in geomechanics in this paper we propose and describe a parallel implementation of a block preconditioner for the solution of saddle point linear systems arising from finite element fe discretization of 3d coupled consolidation problems the mixed constraint preconditioner developed in l bergamaschi m ferronato g gambolati mixed constraint preconditioners for the solution to fe coupled consolidation equations j comput phys 227 23 2008 9885 9897 is combined with the parallel fsai preconditioner which is used here to approximate the inverses of both the structural 1 1 block and an appropriate schur complement matrix the resulting preconditioner proves effective in the acceleration of the bicgstab iterative solver numerical results on a number of test cases of size up to 2x10 6 unknowns and 1 2x10 8 nonzeros show the perfect scalability of the overall code up to 256 processors

simultaneous inversion of the fractional order and the space dependent source term for the time fractional diffusion equation abstract in this paper a simultaneous identification problem of the spacewise source term and the fractional order for a time fractional diffusion equation is considered firstly under some assumption and with two different kinds of observation data for one dimensional and two dimensional time fractional diffusion equation the unique results of the inverse problem are proven by the laplace transformation method and analytic continuation technique then the inverse problems are transformed into tikhonov type optimization problems the existence of optimal solutions to the tikhonov functional is proven finally we adopt an alternating minimization algorithm to solve the optimization problems the efficiency and stability of the inversion algorithm are tested by several one and two dimensional examples

the fractional diffusion equation in one space and in one time dimension a diffusion equation is solved where the first time derivative is replaced by the fractional time derivative 0 1 the solution is given in closed form in terms of fox functions

robust approximate inverse preconditioning for the conjugate gradient method we present a variant of the ainv factorized sparse approximate inverse algorithm which is applicable to any symmetric positive definite matrix the new preconditioner is breakdown free and when used in conjunction with the conjugate gradient method results in a reliable solver for highly ill conditioned linear systems we also investigate an alternative approach to a stable approximate inverse algorithm based on the idea of diagonally compensated reduction of matrix entries the results of numerical tests on challenging linear systems arising from finite element modeling of elasticity and diffusion problems are presented

initial value boundary value problems for fractional diffusion wave equations and applications to some inverse problems abstract we consider initial value boundary value problems for fractional diffusion wave equation t u x t l u x t where 0 2 where l is a symmetric uniformly elliptic operator with t independent smooth coefficients first we establish the unique existence of the weak solution and the asymptotic behavior as the time t goes to and the proofs are based on the eigenfunction expansions second for 0 1 we apply the eigenfunction expansions and prove i stability in the backward problem in time ii the uniqueness in determining an initial value and iii the uniqueness of solution by the decay rate as t iv stability in an inverse source problem of determining t dependent factor in the source by observation at one point over 0 t

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 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

a robust preconditioner for fluid structure interaction problems abstract two preconditioners are presented for equation systems of strongly coupled fluid structure interaction computations where the structure is modeled by shell elements these preconditioners fall into the general category of incomplete lu factorization the two differ mainly in whether the coefficient matrix is factorized node by node or variable by variable in the variable wise preconditioner a modified schur complement system for pressure is solved approximately with a few iterations using a special preconditioner the efficiencies of the two preconditioners are compared for different finite element formulations of the fluid mechanics part including formulations with supg and pspg stabilizations

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

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

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

high order finite element methods for time fractional partial differential equations the aim of this paper is to develop high order methods for solving time fractional partial differential equations the proposed high order method is based on high order finite element method for space and finite difference method for time optimal convergence rate o dt 2 a n r is proved for the r 1 th order finite element method r 2

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

existence and uniqueness of entire solutions for a reaction diffusion equation abstract we consider entire solutions of u t u xx f u i e solutions that exist for all x t r 2 where f 0 f 1 0 f 0 in particular we are interested in the entire solutions which behave as two opposite wave fronts of positive speed s approaching each other from both sides of the x axis and then annihilating in a finite time in the case f 1 0 we show that such entire solution exists and is unique up to space time translations in the case f 1 0 we derive two families of such entire solutions in the first family one cannot be any space time translation of the other yet all entire solutions in the second family only differ by a space time translation

stochastic collocation algorithms using l_1 minimization for bayesian solution of inverse problems the bayesian methodology has proven to be a convenient framework to solve inverse problems from available data with limited information the main computational challenges in the bayesian solution of inverse problems arise from the need for repeated evaluations of the forward model as required by markov chain monte carlo mcmc methods for posterior sampling in this paper we present an efficient technique for constructing stochastic surrogate models to accelerate the bayesian inference approach for statistical inverse problems the stochastic collocation algorithms using l 1 minimization l1 scm based on generalized polynomial chaos are used to construct a polynomial approximation of the forward solution over the support of the prior distribution this approximation then defines a surrogate posterior probability density that is inexpensive to evaluate a rigorous error analysis shows that the convergence rate of the posterior density is at least twice as fast as the convergence rate of the gpc expa

weighted approximate fekete points sampling for least squares polynomial approximation we propose and analyze a weighted greedy scheme for computing deterministic sample configurations in multidimensional space for performing least squares polynomial approximations on l 2 spaces we

simultaneous identification of three parameters in a time fractional diffusion wave equation by a part of boundary cauchy data abstract this paper is devoted to determine the fractional order the initial flux speed and the boundary neumann data simultaneously in a one dimensional time fractional diffusion wave equation from part boundary cauchy observation data we prove the uniqueness result for this inverse problem by using a new result for the mittag leffler function and laplace transform combining with analytic continuation then we use the iterative regularizing ensemble kalman method in bayesian framework to solve the inverse problem numerically and four numerical examples are provided to show the effectiveness and stability of the proposed algorithm

uncertainty quantification and weak approximation of an elliptic inverse problem we consider the inverse problem of determining the permeability from the pressure in a darcy model of flow in a porous medium mathematically the problem is to find the diffusion coefficient for a linear uniformly elliptic partial differential equation in divergence form in a bounded domain in dimension d le 3 from measurements of the solution in the interior we adopt a bayesian approach to the problem we place a prior random field measure on the log permeability specified through the karhunen loeve expansion of its draws we consider gaussian measures constructed this way and study the regularity of functions drawn from them we also study the lipschitz properties of the observation operator mapping the log permeability to the observations combining these regularity and continuity estimates we show that the posterior measure is well defined on a suitable banach space furthermore the posterior measure is shown to be lipschitz with respect to the data in the hellinger metric giving rise to a form of well posedness of the inverse problem determining the posterior measure given the data solves the problem of uncertainty quantification for this inverse problem in practice the posterior measure must be approximated in a finite dimensional space we quantify the errors incurred by employing a truncated karhunen loeve expansion to represent this meausure in particular we study weak convergence of a general class of locally lipschitz functions of the log permeability and apply this general theory to estimate errors in the posterior mean of the pressure and the pressure covariance under refinement of the finite dimensional karhunen loeve truncation

finite difference spectral approximations for the time fractional diffusion equation in this paper we consider the numerical resolution of a time fractional diffusion equation which is obtained from the standard diffusion equation by replacing the first order time derivative with a fractional derivative of order a with 0 a 1 the main purpose of this work is to construct and analyze stable and high order scheme to efficiently solve the time fractional diffusion equation the proposed method is based on a finite difference scheme in time and legendre spectral methods in space stability and convergence of the method are rigourously established we prove that the full discretization is unconditionally stable and the numerical solution converges to the exact one with order o dt 2 a n m where dt n and m are the time step size polynomial degree and regularity of the exact solution respectively numerical experiments are carried out to support the theoretical claims

ill conditioning of finite element poroelasticity equations abstract the solution to biot s coupled consolidation theory is usually addressed by the finite element fe method thus obtaining a system of first order differential equations which is integrated by the use of an appropriate time marching scheme for small values of the time step the resulting linear system may be severely ill conditioned and hence the solution can prove quite difficult to achieve under such conditions efficient and robust projection solvers based on krylov s subspaces which are usually recommended for non symmetric large size problems can exhibit a very slow convergence rate or even fail the present paper investigates the correlation between the ill conditioning of fe poroelasticity equations and the time integration step t an empirical relation is provided for a lower bound tcrit of t below which ill conditioning may suddenly occur the critical time step is larger for soft and low permeable porous media discretized on coarser grids a limiting value for the rock stiffness is found such that for stiffer systems there is no ill conditioning irrespective of t however small as is also shown by several numerical examples finally the definition of a different tcrit as suggested by other authors is reviewed and discussed

determination of the initial data in a time fractional diffusion wave problem by a final time data abstract this paper is to determine an initial value for a time fractional diffusion wave equation from the final time data the regularity of the weak solution for the direct problem is improved the existence and uniqueness of a weak solution for the adjoint problem are proved by using the fourier method the tikhonov regularization method is applied for finding a stable approximate solution in order to get the minimizer of the tikhonov regularization functional we propose a conjugate gradient algorithm based on the strict deductions of the sensitive problem and the adjoint problem four numerical examples in one dimensional and two dimensional cases are provided to show the effectiveness and stability of the proposed algorithm

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

the backward problem for a time fractional diffusion wave equation in a bounded domain abstract this paper is devoted to solve the backward problem for a time fractional diffusion wave equation in a bounded domain based on the series expression of the solution for the direct problem the backward problem for searching the initial data is converted into solving the fredholm integral equation of the first kind the existence uniqueness and conditional stability for the backward problem are investigated we use the tikhonov regularization method to deal with the integral equation and obtain the series expression of the regularized solution for the backward problem furthermore the convergence rate for the regularized solution can be proved by using an a priori regularization parameter choice rule and an a posteriori regularization parameter choice rule numerical results for five examples in one dimensional case and two dimensional case show that the proposed method is efficient and stable

a specialized interior point algorithm for multicommodity network flows despite the efficiency shown by interior point methods in large scale linear programming they usually perform poorly when applied to multicommodity flow problems the new specialized interior point algorithm presented here overcomes this drawback this specialization uses both a preconditioned conjugate gradient solver and a sparse cholesky factorization to solve a linear system of equations at each iteration of the algorithm the ad hoc preconditioner developed by exploiting the structure of the problem is instrumental in ensuring the efficiency of the method an implementation of the algorithm is compared to state of the art packages for multicommodity flows the computational experiments were carried out using an extensive set of test problems with sizes of up to 700 000 variables and 150 000 constraints the results show the effectiveness of the algorithm

approximation of bayesian inverse problems for pdes inverse problems are often ill posed with solutions that depend sensitively on data in any numerical approach to the solution of such problems regularization of some form is needed to counteract the resulting instability this paper is based on an approach to regularization employing a bayesian formulation of the problem which leads to a notion of well posedness for inverse problems at the level of probability measures the stability which results from this well posedness may be used as the basis for quantifying the approximation in finite dimensional spaces of inverse problems for functions this paper contains a theory which utilizes this stability property to estimate the distance between the true and approximate posterior distributions in the hellinger metric in terms of error estimates for approximation of the underlying forward problem this is potentially useful as it allows for the transfer of estimates from the numerical analysis of forward problems into estimates for the solution of the related inverse problem it is noteworthy that when the prior is a gaussian random field model controlling differences in the hellinger metric leads to control on the differences between expected values of polynomially bounded functions and operators including the mean and covariance operator the ideas are applied to some non gaussian inverse problems where the goal is determination of the initial condition for the stokes or navier stokes equation from lagrangian and eulerian observations respectively

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 limit case in nonlinear diffusion on considere le probleme suivant u t u 2 u x x si t 0 x r u 0 x u 0 x si x r cette equation est un modele de diffusion rapide en conduction de chaleur non lineaire on suppose que u 0 x est une fonction reelle localement integrable et non negative on determine des conditions sur u 0 qui assurent l existence d une solution u t x 0 dans un intervalle de temps

a sparse approximate inverse preconditioner for nonsymmetric linear systems this paper is concerned with a new approach to preconditioning for large sparse linear systems a procedure for computing an incomplete factorization of the inverse of a nonsymmetric matrix is developed and the resulting factorized sparse approximate inverse is used as an explicit preconditioner for conjugate gradient type methods some theoretical properties of the preconditioner are discussed and numerical experiments on test matrices from the harwell boeing collection and from tim davis s collection are presented our results indicate that the new preconditioner is cheaper to construct than other approximate inverse preconditioners furthermore the new technique insures convergence rates of the preconditioned iteration which are comparable with those obtained with standard implicit preconditioners

rmcp relaxed mixed constraint preconditioners for saddle point linear systems arising in geomechanics abstract 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 a relaxed mixed constraint preconditioner rmcp which is a generalization based on a parameter of the mixed constraint preconditioner mcp developed in 7 choice of optimal is driven by the spectral distribution of suitable symmetric positive definite spd matrices numerical tests performed on realistic 3d problems reveal that rmcp accelerates krylov subspace solvers by a factor up to three with respect to mcp

existence and nonexistence of curved front solution of a biological equation abstract this paper deals with the existence of curved front solution of a partial differential equation coming from a mathematical model of stroke the equation is of reaction diffusion type in a cylinder of radius r and of diffusion and absorption type outside of the cylinder we prove the nonexistence of a travelling front when r is small enough and the existence if r is large enough using a recent energy method we construct the travelling front as the limit in time of a solution with a well chosen initial condition in a travelling referential

a partially parallel in time fixed stress splitting method for biot s consolidation model in this work we study the parallel in time iterative solution of coupled flow and geomechanics in porous media modelled by a two field formulation of biot s equations in particular we propose a new version of the fixed stress splitting method which has been widely used as solution method of these problems this new approach forgets about the sequential nature of the temporal variable and considers the time direction as a further direction for parallelization the method is partially parallel in time we present a rigorous convergence analysis of the method and numerical experiments to demonstrate the robust behaviour of the algorithm

a phase plane discussion of convergence to travelling fronts for nonlinear diffusion the paper is concerned with the asymptotic behavior as t of solutions u x t of the equation u t u xx f left u right 0 x in left infty infty right in the case f 0 f 1 0 with f u non positive for u 0 sufficiently close to zero and f u non negative for u 1 sufficiently close to 1 this guarantees the uniqueness but not the existence of a travelling front solution u u x ct u 0 u and it is shown in essence that solutions with monotonic initial data converge to a translate of this travelling front if it exists and to a stacked combination of travelling fronts if it does not the approach is to use the monotonicity to take u and t as independent variables and p ux as the dependent variable and to apply ideas of sub and super solutions to the diffusion equation for p

performance and analysis of saddle point preconditioners for the discrete steady state navier stokes equations we examine the convergence characteristics of iterative methods based on a new preconditioning operator for solving the linear systems arising from discretization and linearization of the steady state navier stokes equations with a combination of analytic and empirical results we study the effects of fundamental parameters on convergence we demonstrate that the preconditioned problem has an eigenvalue distribution consisting of a tightly clustered set together with a small number of outliers the structure of these distributions is independent of the discretization mesh size but the cardinality of the set of outliers increases slowly as the viscosity becomes smaller these characteristics are directly correlated with the convergence properties of iterative solvers

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

heteroclinic travelling waves of gradient diffusion systems we establish existence of travelling waves to the gradient system ut uzz w u connecting two minima of w when u r 0 r n that is we establish existence of a pair u c c 2 r n 0 satisfying uxx w u c ux u a where a are local minima of the potential w c 2 r n with w a w a 0 and n 1 our method is variational and based on the mini mization of the functional ec u r r n 1 2 ux 2 w u o e cx dx in the appro priate space setup following alikakos fusco a f we introduce an artificial constraint to restore compactness and force the desired asymptotic behavior which we later remove we provide variational characterizations of the travel ling wave and the speed in particular we show that ec u 0