existence and regularity of invariant measures for the three dimensional stochastic primitive equations we establish the continuity of the markovian semigroup associated with strong solutions of the stochastic 3d primitive equations and prove the existence of an invariant measure the proof is based on new moment bounds for strong solutions the invariant measure is supported on strong solutions and is furthermore shown to have higher regularity properties

superconvergence analysis of nonconforming fem for nonlinear time dependent thermistor problem abstract in this paper we study the superconvergence analysis of nonlinear time dependent thermistor problem with the well known nonconforming element i e the extension of the rotated bilinear element denoted e q 1 r o t for the semi discrete and a linearized backward euler fully discrete schemes the superclose and superconvergent estimates about the related variables in broken h1 norm are derived with the help of the rigorous analysis together with the special properties of this element mean value technique and interpolated post processing approach finally a numerical experiment is carried out to confirm the theoretical analysis

ergodicity of the 2d navier stokes equations with random forcing we consider the navier stokes equation on a two dimensional torus with a random force acting at discrete times and analytic in space for arbitrarily small viscosity coefficient we prove the existence and uniqueness of the invariant measure for this system as well as exponential mixing in time

ergodicity for the navier stokes equation with degenerate random forcing finite dimensional approximation we study galerkin truncations of the two dimensional navier stokes equation under degenerate large scale stochastic forcing we identify the minimal set of modes that has to be forced in order for the system to be ergodic our results rely heavily on the structure of the nonlinearity c 2001 john wiley sons inc

stochastic dissipative pde s and gibbs measures we study a class of dissipative nonlinear pde s forced by a random force omega t x with the space variable x varying in a bounded domain the class contains the 2d navier stokes equations under periodic or dirichlet boundary conditions and the forces we consider are those common in statistical hydrodynamics they are random fields smooth in t and stationary short correlated in time t in this paper we confine ourselves to kick forces of the form where the k s are smooth bounded identically distributed random fields the equation in question defines a markov chain in an appropriately chosen phase space a subset of a function space that contains the zero function and is invariant for the random flow of the equation concerning this markov chain we prove the following main result see theorem 2 2 the markov chain has a unique invariant measure to prove this theorem we present a construction assigning to any invariant measure a gibbs measure for a 1d system with compact phase space and apply a version of ruelle perron frobenius uniqueness theorem to the corresponding gibbs system we also discuss ergodic properties of the invariant measure and corresponding properties of the original randomly forced pde

finite element approximation to nonlinear coupled thermal problem a nonlinear coupled elliptic system modelling a large class of engineering problems was discussed in a f d loula j zhu finite element analysis of a coupled nonlinear system comp appl math 20 3 2001 321 339 j zhu a f d loula mixed finite element analysis of a thermally nonlinear coupled problem numer methods partial differential equations 22 1 2006 180 196 the convergence analysis of iterative finite element approximation to the solution was done under an assumption of small solution or source data which guarantees the uniqueness of the nonlinear coupled system generally a nonlinear system may have multiple solutions in this work the regularity of the weak solutions is further studied the nonlinear finite element approximations to the nonsingular solutions are then proposed and analyzed finally the optimal order error estimates in h 1 norm and l 2 norm as well as in w 1 p norm and l p norm are obtained

a ϕ formulation of a mathematical model for the induction hardening process with a nonlinear law for the magnetic field abstract we derive and analyze a mathematical model for induction hardening we assume a nonlinear relation between the magnetic field and the magnetic induction field for the electromagnetic part we use the vector scalar potential formulation the coupling between the electromagnetic and the thermal part is provided through the temperature dependent electric conductivity and the joule heating term the most crucial element considering the mathematical analysis of the model it acts as a source of heat in the thermal part and leads to the increase in temperature therefore in order to be able to control it we apply a truncation function using rothe s method we prove the existence of a global solution to the whole system the nonlinearity in the electromagnetic part is handled by the theory of monotone operators to supplement our theoretical results we provide a numerical simulation using real physical constants

stability and convergence of fully discrete galerkin fems for the nonlinear thermistor equations in a nonconvex polygon in this paper we establish the unconditional stability and optimal error estimates of a linearized backward euler galerkin finite element method fem for the time dependent nonlinear thermistor equations in a two dimensional nonconvex polygon due to the nonlinearity of the equations and the non smoothness of the solution in a nonconvex polygon the analysis is not straightforward while most previous efforts for problems in nonconvex polygons mainly focused on linear models our theoretical analysis is based on an error splitting proposed in 30 31 together with rigorous regularity analysis of the nonlinear thermistor equations and the corresponding iterated time discrete elliptic system in a nonconvex polygon with the proved regularity we establish the stability in l infty l infty l l and the convergence in l infty l 2 l l2 for the fully discrete finite element solution without any restriction on the time step size the approach used in this paper may also be applied to other nonlinear parabolic systems in nonconvex polygons numerical results confirm our theoretical analysis and show clearly that no time step condition is needed

spectral gaps in wasserstein distances and the 2d stochastic navier stokes equations we develop a general method to prove the existence of spectral gaps for markov semigroups on banach spaces unlike most previous work the type of norm we consider for this analysis is neither a weighted supremum norm nor an lp type norm but involves the derivative of the observable as well and hence can be seen as a type of 1 wasserstein distance this turns out to be a suitable approach for infinite dimensional spaces where the usual harris or doeblin conditions which are geared toward total variation convergence often fail to hold in the first part of this paper we consider semigroups that have uniform behavior which one can view its the analog of doeblin s condition we then proceed to study situations where the behavior is not so uniform but the system has a suitable lyapunov structure leading to a type of harris condition we finally show that the latter condition is satisfied by the two dimensional stochastic navier stokes equations even in situations where the forcing is extremely de generate using the convergence result we show that the stochastic navier stokes equations invariant measures depend continuously on the viscosity and the structure of the forcing

superconvergence analysis of finite element method for time fractional thermistor problem abstract in this paper the superclose and superconvergence analysis of the nonlinear time fractional thermistor problem are investigated by bilinear finite element method fem for a fully discrete scheme in which the caputo derivative is approximated by the classical l 1 method by dealing with the error estimates in the spatial direction rigorously which are one order higher than the traditional fems the superclose estimates in h 1 norm are obtained for the corresponding variables based on the special properties of this element together with mean value technique subsequently the global superconvergence results are derived by employing the interpolation postprocessing approach finally a numerical experiment is carried out to confirm the theoretical analysis

a finite element model for the time dependent joule heating problem we study a spatially semidiscrete and a completely discrete finite element model for a nonlinear system consisting of an elliptic and a parabolic partial differential equation describing the electric heating of a conducting body we prove error bounds of optimal order under minimal regularity assumptions when the number of spatial variables d 3 we establish the existence of solutions with the required regularity over arbitrarily long intervals of time when d 2

ergodicity of 2d navier stokes equations with random forcing and large viscosity the stochastically forced two dimensional incompressable navier stokes equations are shown to possess an unique invariant measure if the viscosity is taken large enough this result follows from a stronger result showing that at high viscosity there is a unique stationary solution which attracts solutions started from arbitrary initial conditions that is to say the system has a trivial random attractor along the way results controling the expectation and averaging time of the energy and enstrophy are given

induction hardening of steel with restrained joule heating and nonlinear law for magnetic induction field we study a coupled system of maxwell s equations with nonlinear heat equation we start with derivation of a mathematical model assuming a nonlinear dependency of magnetic field h on magnetic induction b moreover the electric conductivity is supposed to be temperature reliant the coupling between the thermal and electromagnetic equations is provided via the joule heating term and next we use time discretization based on the rothe s method to provide energy estimates for discretized system we prove the existence of a weak solution u b to this coupled system with controlled joule heating term

postprocessing the finite element method for semilinear parabolic problems in this paper we consider postprocessing of the finite element method for semilinear parabolic problems the postprocessing amounts to solving a linear elliptic problem on a finer grid or higher order space once the time integration on the coarser mesh is completed the convergence rate is increased at almost no additional computational cost this procedure was introduced and analyzed in garcia archilla and titi siam j numer anal 37 2000 pp 470 499 we extend the analysis to the fully discrete case and prove error estimates for both space and time discretization the analysis is based on error estimates for the approximation of time derivatives by difference quotients

variational formulation of a model free boundary problem the purpose of this work is to present an error analysis of the numerical approximation by a finite element method of a free surface problem the analysis has been done in an abstract model which has many of the features of a free surface problem for a viscous liquid we study in this paper how the numerical approximation of the free boundary affects the approximation of the other variables of the problem and vice versa we present the numerical analysis of a free boundary problem that is intended to incorporate many of the difficulties found in a class of models of fluid flow phenomena with free surfaces one such phenomenon which motivates the current work is the flow of a liquid constrained only partly by a container that is in which a part of the boundary of the domain filled by the liquid is an interface with another liquid of much smaller density and for which surface tension plays a significant role in determining the shape of the free surface one model for the behavior of such liquids is based on the assumption that the surface tension between the two liquids is proportional to the curvature of the free surface the constant of proportionality is a physical property of the two fluids this model has been studied extensively in recently years both experimentally cf jean and pritchard 15 and pritchard 19 theoretically cf allain 2 beale 4 bemelmans 5 jean 14 pukhnachov 20 and solonnikov 27 asymptotically keller and miksis 16 and computationally cf cuvelier 10 ryskin and leal 23 and saito and scriven 24 our purpose here is to establish a framework for the analysis of convergence properties of the computational techniques being used the only previous work that we are aware of in this direction is by nitsche 18 in the first section of the paper we define our model problem in classical terms in the second section we construct a variational formulation for the problem that has two new features one is that it allows the existence of a solution to be proved with weaker assumptions on the data than has been possible received february 26 1990 revised november 20 1990 1980 mathematics subject classification 1985 revision primary 65n30

error estimates for spatially discrete approximations of semilinear parabolic equations with nonsmooth initial data on considere les approximations spatialement discretes continues par rapport au temps des problemes mixtes par la methode des elements finis de galerkin pour les equations paraboliques semi lineaires avec des donnees initiales irregulieres ou incompatibles on etablit des estimations d erreur

gibbsian dynamics and ergodicity for the stochastically forced navier stokes equation we study stationary measures for the two dimensional navier stokes equation with periodic boundary condition and random forcing we prove uniqueness of the stationary measure under the condition that all determining modes are forced the main idea behind the proof is to study the gibbsian dynamics of the low modes obtained by representing the high modes as functionals of the time history of the low modes

finite element approximation of the cahn hilliard equation with concentration dependent mobility we consider the cahn hilliard equation with a logarithmic free energy and non degenerate concentration dependent mobility in particular we prove that there exists a unique solution for sufficiently smooth initial data further we prove an error bound for a fully practical piecewise linear finite element approximation in one and two space dimensions finally some numerical experiments are presented

unconditionally optimal error estimates of a crank nicolson galerkin method for the nonlinear thermistor equations this paper focuses on unconditionally optimal error analysis of an uncoupled and linearized crank nicolson galerkin finite element method for the time dependent nonlinear thermistor equations in d dimensional space d 2 3 in our analysis we split the error function into two parts one from the spatial discretization and one from the temporal discretization by introducing a corresponding time discrete elliptic system we present a rigorous analysis for the regularity of the solution of the time discrete system and error estimates of the time discretization with these estimates and the proved regularity optimal error estimates of the fully discrete crank nicolson galerkin method are obtained unconditionally numerical results confirm our analysis and show the efficiency of the method

on ergodicity of some markov processes we formulate a criterion for the existence and uniqueness of an invariant measure for a markov process taking values in a polish phase space in addition weak ergodicity that is the weak convergence of the ergodic averages of the laws of the process starting from any initial distribution is established the principal assumptions are the existence of a lower bound for the ergodic averages of the transition probability function and its local uniform continuity the latter is called the e property the general result is applied to solutions of some stochastic evolution equations in hilbert spaces as an example we consider an evolution equation whose solution describes the lagrangian observations of the velocity field in the passive tracer model the weak mean ergodicity of the corresponding invariant measure is used to derive the law of large numbers for the trajectory of a tracer

exponential convergence for the stochastically forced navier stokes equations and other partially dissipative dynamics we prove that the two dimensional navier stokes equations possess an exponentially attracting invariant measure this result is in fact the consequence of a more general harris like ergodic theorem applicable to many dissipative stochastic pdes and stochastic processes with memory a simple iterated map example is also presented to help build intuition and showcase the central ideas in a less encumbered setting to analyze the iterated map a general doeblin like theorem is proven one of the main features of this paper is the novel coupling construction used to examine the ergodic theory of the non markovian processes

unconditional convergence and optimal error estimates of a galerkin mixed fem for incompressible miscible flow in porous media in this paper we study the unconditional convergence and error estimates of a galerkin mixed fem with the linearized semi implicit euler scheme for the equations of incompressible miscible flow in porous media we prove that the optimal l 2 error estimates hold without any time step convergence conditions while all previous works require certain time step restrictions our theoretical results provide a new understanding on commonly used linearized schemes the proof is based on a splitting of the error into two parts the error from the time discretization of the pdes and the error from the finite element discretization of corresponding time discrete pdes the approach used in this paper can be applied to more general nonlinear parabolic systems and many other linearized semi implicit time discretizations

the small scales of the stochastic navier stokes equations under rough forcing we prove that the small scale structures of the stochastically forced navier stokes equations approach those of the naturally associated ornstein uhlenbeck process as the scales get smaller precisely we prove that the rescaled kth spatial fourier mode converges weakly on path space to an associated ornstein uhlenbeck process as k in addition we prove that the navier stokes equations and the naturally associated ornstein uhlenbeck process induce equivalent transition densities if the viscosity is replaced with hyper viscosity this gives a simple proof of unique ergodicity for the hyperviscous navier stokes system we show how different strengthened hyperviscosity produce varying levels of equivalence

a model for two coupled turbulent fluids part ii numerical analysis of a spectral discretization we consider a system of equations that models the stationary flow of two immiscible turbulent fluids on adjacent subdomains the equations are coupled by nonlinear boundary conditions on the interface which is here a fixed given surface we propose a spectral discretization of this problem and perform its numerical analysis the convergence of the method is proven in the two dimensional case together with optimal error estimates

ergodicity of the 2 d navier stokes equation under random perturbations a 2 dimensional navier stokes equation perturbed by a sufficiently distributed white noise is considered existence of invariant measures is known from previous works the aim is to prove uniqueness of the invariant measures strong law of large numbers and convergence to equilibrium

complexified gravity in noncommutative spaces the presence of a constant background antisymmetric tensor for open strings or d branes forces the space time coordinates to be noncommutative this effect is equivalent to replacing ordinary products in the effective theory by the deformed star product an immediate consequence of this is that all fields get complexified the only possible noncommutative yang mills theory is the one with u n gauge symmetry by applying this idea to gravity one discovers that the metric becomes complex we show in this article that this procedure is completely consistent and one can obtain complexified gravity by gauging the symmetry u 1 d 1 instead of the usual so 1 d 1 the final theory depends on a hermitian tensor containing both the symmetric metric and antisymmetric tensor in contrast to other theories of nonsymmetric gravity the action is both unique and gauge invariant the results are then generalized to noncommutative spaces

mixed variational formulation and numerical analysis of thermally coupled nonlinear darcy flows a mixed variational formulation and its finite element approximate procedure combined with a fixed point algorithm is presented for solving a thermally coupled nonlinear darcy flow problem existence and uniqueness of the solution to the nonlinear mixed variational formulation are established a more general result on uniqueness of the continuous problem is presented numerical analysis of the finite element approximation with the corresponding error estimates is given numerical results are presented confirming the expected rates of convergence and illustrating the influence of the exponent of the nonlinear power law constitutive equation

thermally coupled quasi newtonian flows analysis and computation in this paper we consider an incompressible quasi newtonian flow with a temperature dependent viscosity obeying a power law and the thermal balance includes viscous heating some mathematical results such as the existence and uniqueness are established finite element approximation based on an iterative solution scheme is proposed and convergence analysis is presented

mixed discontinuous galerkin analysis of thermally nonlinear coupled problem abstract a stabilized mixed discontinuous galerkin smdg method based on brezzi hughes marini masud f brezzi t j r hughes l d marini a masud mixed discontnuous galerkin methods for darcy flow j sci comput 22 2005 119 145 is proposed to solve a thermally coupled nonlinear elliptic system modeling a large class of engineering problems a fixed point algorithm is adopted to solve the nonlinear systems convergence analysis and error estimates are presented for equal order linear or bilinear discontinuous lagrangian finite element interpolations for all fields numerical results are presented confirming the predicted convergence rates and illustrating the performance of the proposed formulation solving problems with globally stable and blowing up solutions