rates of convergence for discretizations of the stochastic incompressible navier stokes equations we show strong convergence with rates for an implicit time discretization a semi implicit time discretization and a related finite element based space time discretization of the incompressible navier stokes equations with multiplicative noise in two space dimensions we use higher moments of computed iterates to optimally bound the error on a subset omega kappa of the sample space omega where corresponding paths are bounded in a proper function space and mathbb p omega kappa to 1 holds for vanishing discretization parameters this implies convergence in probability with rates and motivates a practicable acception rejection criterion to overcome possible pathwise explosion behavior caused by the nonlinearity it turns out that it is the interaction of lagrange multipliers with the stochastic forcing in the scheme which limits the accuracy of general discretely lbb stable space discretizations and strategies to overcome this problem are proposed

an accelerated exponential time integrator for semi linear stochastic strongly damped wave equation with additive noise abstract this paper is concerned with the strong approximation of a semi linear stochastic wave equation with strong damping driven by additive noise based on a spatial discretization performed by a spectral galerkin method we introduce a kind of accelerated exponential time integrator involving linear functionals of the noise under appropriate assumptions we provide error bounds for the proposed full discrete scheme it is shown that the scheme achieves higher strong order in time direction than the order of temporal regularity of the underlying problem which allows for higher convergence rate than usual time stepping schemes particularly for the space time white noise case in two or three spatial dimensions the scheme still exhibits a good convergence performance another striking finding is that even for the velocity with low regularity the scheme always promises first order strong convergence in time numerical examples are finally reported to confirm our theoretical findings

higher order pathwise numerical approximations of spdes with additive noise in this article the pathwise numerical approximation of semilinear parabolic stochastic partial differential equations spdes driven by additive noise is considered a new numerical scheme for the time and space discretization of such spdes is proposed and this scheme is shown to converge for such spdes faster than standard numerical schemes such as the linear implicit euler scheme or the linear implicit crank nicholson scheme the suggested scheme takes advantage of the smoothing effect of the dominant linear operator and of two linear functionals of the noise process of the spde the abstract setting under which the scheme is analyzed includes spdes driven by fractional brownian motions too

full discretization of semilinear stochastic wave equations driven by multiplicative noise a fully discrete approximation of the semilinear stochastic wave equation driven by multiplicative noise is presented a standard linear finite element approximation is used in space and a stochastic trigonometric method is used for the temporal approximation this explicit time integrator allows for mean square error bounds independent of the space discretization and thus does not suffer from a step size restriction as in the often used stormer verlet leapfrog scheme furthermore it satisfies an almost trace formula i e a linear drift of the expected value of the energy of the problem numerical experiments are presented and confirm the theoretical results

the approximation of a crank nicolson scheme for the stochastic navier stokes equations in this paper we prove the convergence of stochastic navier stokes equations driven by white noise a linearized version of the implicit crank nicolson scheme is considered for the approximation of the solutions to the n s equations the noise is defined as the distributional derivative of a wiener process and approximated by using the generalized l 2 projection operator optimal strong convergence error estimates in the l 2 norm are obtained

convergence of finite element solutions of stochastic partial integro differential equations driven by white noise numerical approximation of a stochastic partial integro differential equation driven by a space time white noise is studied by truncating a series representation of the noise with finite element method for spatial discretization and convolution quadrature for time discretization sharp order convergence of the numerical solutions is proved up to a logarithmic factor numerical examples are provided to support the theoretical analysis

optimal strong convergence rates of numerical methods for semilinear parabolic spde driven by gaussian noise and poisson random measure abstract this paper deals with the numerical approximation of semilinear parabolic stochastic partial differential equation spde driven simultaneously by gaussian noise and poisson random measure more realistic in modeling real world phenomena the spde is discretized in space with the standard finite element method and in time with the linear implicit euler method or an exponential integrator more efficient and stable for stiff problems we prove the strong convergence of the fully discrete schemes toward the mild solution the results reveal how convergence orders depend on the regularity of the noise and the initial data in addition we exceed the classical orders 1 2 in time and 1 in space achieved in the literature when dealing with spde driven by poisson measure with less regularity assumptions on the nonlinear drift function in particular for trace class multiplicative gaussian noise we achieve convergence order o h 2 t 1 2 for additive trace class gaussian noise and an appropriate jump function we achieve convergence order o h 2 t numerical experiments to sustain the theoretical results are provided

weak approximation of stochastic partial differential equations the nonlinear case we study the error of the euler scheme applied to a stochastic partial dierential equation we prove that as it is often the case the weak order of convergence is twice the strong order a key ingredient in our proof is malliavin calculus which enables us to get rid of the irregular terms of the error we apply our method to the case a semilinear stochastic heat equation driven by a space time white noise

finite element approximation of the linear stochastic wave equation with additive noise semidiscrete finite element approximation of the linear stochastic wave equation lswe with additive noise is studied in a semigroup framework optimal error estimates for the deterministic problem are obtained under minimal regularity assumptions these are used to prove strong convergence estimates for the stochastic problem the theory presented here applies to multidimensional domains and spatially correlated noise numerical examples illustrate the theory

a posteriori error estimates for finite element approximation of unsteady incompressible stochastic navier stokes equations in this paper we consider the unsteady incompressible stochastic navier stokes equations containing a noise part the purpose of this paper is to derive a posteriori error estimates for the finite element approximation of stochastic equations applied the weighted clement type interpolator we obtain the posteriori error upper and lower bounds for the semidicretization scheme and the full backward euler discretization scheme

weak order for the discretization of the stochastic heat equation in this paper we study the approximation of the distribution of x t hilbert valued stochastic process solution of a linear parabolic stochastic partial differential equation written in an abstract form as dx t ax t dt q 1 2 d w t quad x 0 x in h quad t in 0 t driven by a gaussian space time noise whose covariance operator q is given we assume that a alpha is a finite trace operator for some alpha 0 and that q is bounded from h into d a beta for some beta geq 0 it is not required to be nuclear or to commute with a the discretization is achieved thanks to finite element methods in space parameter h 0 and implicit euler schemes in time parameter delta t t n we define a discrete solution x n h and for suitable functions varphi defined on h we show that e varphi x n h e varphi x t o h 2 gamma delta t gamma noindent where gamma 1 alpha beta let us note that as in the finite dimensional case the rate of convergence is twice the one for pathwise approximations

pathwise holder convergence of the implicit linear euler scheme for semi linear spdes with multiplicative noise in this article we prove pathwise holder convergence with optimal rates of the implicit euler scheme for the abstract stochastic cauchy problem begin aligned left begin aligned du t au t dt f t u t dt g t u t dw h t quad t in 0 t u 0 x 0 end aligned right end aligned here a is the generator of an analytic c 0 semigroup on a umd banach space x w h is a cylindrical brownian motion in a hilbert space h and the functions f 0 t times x rightarrow x theta f and g 0 t times x rightarrow fancyscript l h x theta g satisfy appropriate local lipschitz conditions the results are applied to a class of second order parabolic spdes driven by multiplicative space time white noise

the 3 colorability problem on graphs with maximum degree four the 3 colorability problem is known to be np complete in the class of graphs with maximum degree four on the other hand due to the celebrated theorem of brooks the problem has a polynomial time solution for graphs with maximum degree three to make the complexity gap more precise we study a family of intermediate graph classes between these two extremes and classify all of them according to the computational complexity of the problem in particular we generalize brooks s theorem in the case of 3 colorability to a larger class by showing that every connected graph in that class is 3 colorable unless it is a complete graph on four vertices

strong and weak convergence order of finite element methods for stochastic pdes with spatial white noise we investigate the strong and weak convergence order of piecewise linear finite element methods for a class of semilinear elliptic equations with additive spatial white noise using a spectral truncation of white noise taking the number of modes for spectral approximation of the white noise at the order of 1 h d 1 hd we show that the strong convergence order of the finite element approximation is h 2 d 2 h2 d 2 where h is the element size and d le 3 d 3 is the dimension moreover we show that the weak convergence order is h min 4 d 2 hmin 4 d 2 we then consider a fourth order equation and an advection reaction equation and show that a spectral approximation of the white noise can lead to higher convergence order in both strong and weak sense when the solutions are smooth numerical results confirm our prediction for one and two dimensional elliptic problems

strong convergence rate of splitting schemes for stochastic nonlinear schrodinger equations abstract in this paper we show that solutions of stochastic nonlinear schrodinger nls equations can be approximated by solutions of coupled splitting systems based on these systems we propose a new kind of fully discrete splitting schemes which possess algebraic strong convergence rates for stochastic nls equations key ingredients of our approach are using the exponential integrability and stability of the corresponding splitting systems and numerical approximations in particular under very mild conditions we derive the optimal strong convergence rate o n 2 1 2 of the spectral splitting crank nicolson scheme where n and denote the dimension of the approximate space and the time step size respectively

weak convergence analysis of the linear implicit euler method for semilinear stochastic partial differential equations with additive noise abstract in this paper we analyze the weak error of a semi discretization in time by the linear implicit euler method for semilinear stochastic partial differential equations spdes with additive noise the main result reveals how the weak order depends on the regularity of noise and that the order of weak convergence is twice that of strong convergence in particular the linear implicit euler method for spdes driven by trace class noise achieves an almost optimal order 1 for arbitrarily small 0

error estimates for semidiscrete finite element approximations of linear and semilinear parabolic equations under minimal regularity assumptions semidiscrete finite element error estimates for linear parabolic equations are derived under minimal regularity with the help of l2 projectors then analogous minimal regularity semidiscrete error estimates for semilinear parabolic equations are derived

galerkin finite element methods for stochastic parabolic partial differential equations we study the finite element method for stochastic parabolic partial differential equations driven by nuclear or space time white noise in the multidimensional case the discretization with respect to space is done by piecewise linear finite elements and in time we apply the backward euler method the noise is approximated by using the generalized l2 projection operator optimal strong convergence error estimates in the l2 and dot h 1 norms with respect to the spatial variable are obtained the proof is based on appropriate nonsmooth data error estimates for the corresponding deterministic parabolic problem the computational analysis and numerical example are given

cubature on wiener space in infinite dimension we prove a stochastic taylor expansion for stochastic partial differential equations spdes and apply this result to obtain cubature methods i e high order weak approximation schemes for spdes

pathwise convergence of an efficient scheme for spdes with non globally lipschitz nonlinearity abstract this paper aims to extend the scheme proposed in jentzen et al 2011 for stochastic partial differential equations spdes with global lipschitz coefficients to non global lipschitz coefficients we investigate the pathwise convergence of the scheme for a class of semilinear parabolic spdes with non globally lipschitz nonlinearity we show first that the scheme is convergent uniformly in time and space under lipschitz assumption on the nonlinearity of the spde then we obtain the convergence in the case of non globally lipschitz nonlinearity via a localization technique compared to the scheme introduced in jentzen 2009 for spdes under non global lipschitz coefficients the scheme considered in this paper is simpler in the sense that the former uses two linear functionals of the noise while the latter uses one

a hyperbolicity preserving discontinuous stochastic galerkin scheme for uncertain hyperbolic systems of equations abstract intrusive uncertainty quantification methods such as stochastic galerkin are gaining popularity whereas the classical stochastic galerkin approach is not ensured to preserve hyperbolicity of the underlying hyperbolic system we apply a modification of this method that uses a slope limiter to retain admissible solutions of the system while providing high order approximations in the physical and stochastic space this is done using a spatial discontinuous galerkin scheme and a multi element stochastic galerkin ansatz in the random space we analyze the convergence of the resulting scheme and apply it to the compressible euler equations with various uncertain initial states in one and two spatial domains with up to three uncertainties the performance in multiple stochastic dimensions is compared to the non intrusive stochastic collocation method the numerical results underline the strength of our method especially if discontinuities are present in the uncertainty of the solution

convergence of numerical schemes for the solution of parabolic stochastic partial differential equations we consider the numerical solution of the stochastic partial differential equation u t 2 u x 2 u w x t where w is space time white noise using finite differences for this equation gyongy has obtained an estimate of the rate of convergence for a simple scheme based on integrals of w over a rectangular grid we investigate the extent to which this order of convergence can be improved and find that better approximations are possible for the case of additive noise u 1 if we wish to estimate space averages of the solution rather than pointwise estimates or if we are permitted to generate other functionals of the noise but for multiplicative noise u u we show that no such improvements are possible

strong convergence of the linear implicit euler method for the finite element discretization of semilinear spdes driven by multiplicative or additive noise abstract this paper aims to investigate the numerical approximation of a general second order parabolic stochastic partial differential equation spde driven by multiplicative or additive noise the linear operator is not necessary self adjoint so more useful in concrete applications the spde is discretized in space by the finite element method and in time by the linear implicit euler method the corresponding scheme is more stable and efficient to solve stochastic advection dominated reactive transport in porous media this extends the current results in the literature to not necessary self adjoint operator as a challenge key part of the proof does not rely anymore on the spectral decomposition of the linear operator the results reveal how the convergence orders depend on the regularity of the noise and the initial data in particular for multiplicative trace class noise we achieve optimal convergence order o h 2 t 1 2 and for additive trace class noise we achieve optimal convergence order in space and sup optimal convergence order in time of the form o h 2 t 1 for an arbitrarily small 0 numerical experiments to sustain our theoretical results are provided

multidimensional high field limit of the electrostatic vlasov poisson fokker planck system abstract in this paper the high field limit of the vlasov poisson fokker planck system for charged particles is rigorously derived the first result is obtained in any space dimension by using modulated energy techniques it requires the smoothness of the solutions of the limit problem in dimension 2 it is possible to handle more general data by using methods developed for a diagonal defect measures theory the convergence of the concentration of particles is obtained in the space of bounded measures in both cases the limit of the sequence of densities of distribution functions is shown to solve a nonlinear system of partial differential equations which is related to ohm s law

numerical approximation of some linear stochastic partial differential equations driven by special additive noises this paper is concerned with the numerical approximation of some linear stochastic partial differential equations with additive noises a special representation of the noise is considered and it is compared with general representations of noises in the infinite dimensional setting convergence analysis and error estimates are presented for the numerical solution based on the standard finite difference and finite element methods the effects of the noises on the accuracy of the approximations are illustrated results of the numerical experiments are provided

a dynamic solver consistent minimum action method this paper discusses the necessity and strategy to unify the development of a dynamic solver and a minimum action method mam for a spatially extended system when employing the large deviation principle ldp to study the effects of small random perturbations a dynamic solver is used to approximate the unperturbed system and a minimum action method is used to approximate the ldp which corresponds to solving an euler lagrange equation related to but more complicated than the unperturbed system we will clarify possible inconsistencies induced by independent numerical approximations of the unperturbed system and the ldp based on which we propose to define both the dynamic solver and the mam on the same approximation space for spatial discretization the semi discrete ldp can then be regarded as the exact ldp of the semi discrete unperturbed system which is a finite dimensional ode system we achieve this methodology for the two dimensional navier stokes equations using a divergence free approximation space the method developed can be used to study the nonlinear instability of wall bounded parallel shear flows and be generalized straightforwardly to three dimensional cases numerical experiments are presented

numerical analysis of semilinear stochastic evolution equations in banach spaces the solution of stochastic evolution equations generally relies on numerical computation here usually the main idea is to discretize the spde spatially obtaining a system of sdes that can be solved by e g the euler scheme in this paper we investigate the discretization error of semilinear stochastic evolution equations in lp spaces resp banach spaces the space discretization may be done by galerkin approximation for the time discretization we consider the implicit euler the explicit euler scheme and the crank nicholson scheme in the last section we give some examples i e we consider an spdes driven by nuclear wiener noise approximated by wavelets and delay equation approximated by finite differences

spatial approximation of stochastic convolutions we study linear stochastic evolution partial differential equations driven by additive noise we present a general and flexible framework for representing the infinite dimensional wiener process which drives the equation since the eigenfunctions and eigenvalues of the covariance operator of the process are usually not available for computations we propose an expansion in an arbitrary frame we show how to obtain error estimates when the truncated expansion is used in the equation for the stochastic heat and wave equations we combine the truncated expansion with a standard finite element method and derive a priori bounds for the mean square error specializing the frame to biorthogonal wavelets in one variable we show how the hierarchical structure support and cancelation properties of the primal and dual bases lead to near sparsity and can be used to simplify the simulation of the noise and its update when new terms are added to the expansion

a multistage wiener chaos expansion method for stochastic advection diffusion reaction equations using wiener chaos expansion wce we develop numerical algorithms for solving second order linear parabolic stochastic partial differential equations spdes we propose a deterministic wce based algorithm for computing moments of the spde solutions without any use of the monte carlo technique we also compare the proposed deterministic algorithm with two other numerical methods based on the monte carlo technique and demonstrate that the new method is more efficient for highly accurate solutions numerical tests verify that the scheme is of mean square order o frac delta n 2 sqrt n 1 for diffusion and for diffusion reaction spdes with constant or variable coefficients where delta is the time step and n is the wiener chaos order

strong order of convergence of a fully discrete approximation of a linear stochastic volterra type evolution equation in this paper we investigate a discrete approximation in time and in space of a hilbert space valued stochastic process u t t 0 t satisfying a stochastic linear evolution equation with a positive type memory term driven by an additive gaussian noise the equation can be written in an abstract form as du t 0 b t s au s ds dt dw t 0 t u 0 u0 h where w is a q wiener process on h l d and where the main example of b we consider is given by b t t 1 0 0 such that a has finite trace and that q is bounded from h into d a for some real with 1 1 the discretization is achieved via an implicit euler scheme and a laplace transform convolution quadrature in time parameter t t n and a standard continuous finite element method in space parameter h let un h be the discrete solution at t n t we show that e un h u t 1 2 o h t for any 1 1 2 and 1 1 m kovacs department of mathematics and statistics university of otago po box 56 dunedin 9054 new zealand e mail mkovacs maths otago ac nz j printems laboratoire d analyse et de mathematiques appliquees cnrs umr 8050 61 avenue du general de gaulle universite paris est 94010 creteil france e mail printems u pec fr 2 mihaly kovacs jacques printems

a modified semi implicit euler maruyama scheme for finite element discretization of spdes with additive noise abstract we consider the numerical approximation of a general second order semi linear parabolic stochastic partial differential equation spde driven by additive space time noise we introduce a new modified scheme using linear functionals of the noise with the semi implicit euler maruyama method in time and the finite element method in space although extension to finite differences or finite volumes would be possible we prove the convergence in the root mean square l2 norm for a diffusion reaction equation and diffusion advection reaction equation with a large family of lipschitz nonlinear functions we present numerical results for a linear reaction diffusion equation in two dimensions as well as a nonlinear example of two dimensional stochastic advection diffusion reaction equation we observe from both the analysis and numerics that the proposed scheme has better convergence properties than the standard semi implicit euler maruyama method