an adaptive finite element moreau yosida based solver for a coupled cahn hilliard navier stokes system an adaptive a posteriori error estimator based finite element method for the numerical solution of a coupled cahn hilliard navier stokes system with a double obstacle homogenous free interfacial energy density is proposed a semi implicit euler scheme for the time integration is applied which results in a system coupling a quasi stokes or oseen type problem for the fluid flow to a variational inequality for the concentration and the chemical potential according to the cahn hilliard model 16 a moreau yosida regularization is employed which relaxes the constraints contained in the variational inequality and thus enables semi smooth newton solvers with locally superlinear convergence in function space moreover upon discretization this yields a mesh independent method for a fixed relaxation parameter for the finite dimensional approximation of the concentration and the chemical potential piecewise linear and globally continuous finite elements are used and for the numerical approximation of the fluid velocity taylor hood finite elements are employed the paper ends by a report on numerical examples showing the efficiency of the new method

efficient adaptive energy stable schemes for the incompressible cahn hilliard navier stokes phase field models in this paper we develop a fully adaptive energy stable scheme for cahn hilliard navier stokes system which is a phase field model for two phase incompressible flows consisting a cahn hilliard type diffusion equation and a navier stokes equation this scheme which is decoupled and unconditionally energy stable based on stabilization involves adaptive mesh adaptive time and a nonlinear multigrid finite difference method numerical experiments are carried out to validate the scheme for problems with matched density and non matched density and also demonstrate that cpu time can be significantly reduced with our adaptive approach

a phase field model and its numerical approximation for two phase incompressible flows with different densities and viscosities modeling and numerical approximation of two phase incompressible flows with different densities and viscosities are considered a physically consistent phase field model that admits an energy law is proposed and several energy stable efficient and accurate time discretization schemes for the coupled nonlinear phase field model are constructed and analyzed ample numerical experiments are carried out to validate the correctness of these schemes and their accuracy for problems with large density and viscosity ratios

decoupled energy stable schemes for phase field models of two phase incompressible flows in this paper we construct two classes based on stabilization and convex splitting of decoupled unconditionally energy stable schemes for cahn hilliard phase field models of two phase incompressible flows at each time step these schemes require solving only a sequence of elliptic equations including a pressure poisson equation furthermore all of these elliptic equations are linear for the schemes based on stabilization making them the first to the best of the authors knowledge totally decoupled linear unconditionally energy stable schemes for phase field models of two phase incompressible flows thus the schemes constructed in this paper are very efficient and easy to implement

inexact block monotone methods for solving nonlinear elliptic problems abstract the paper deals with numerical solving of semilinear elliptic problems based on the method of upper and lower solutions inexact block monotone iterative methods are constructed where monotone linear systems are solved by the block jacobi or block gauss seidel methods only approximately the inexact block monotone methods combine the quadratic monotone iterative method at outer iterations and the block iterative methods at inner iterations and possess global monotone convergence results of numerical experiments implemented in the framework of an inexact newton method are presented where iteration counts and cpu times of the inexact block monotone methods are compared with the block monotone iterative methods whose convergence rate is linear

quasi incompressible cahn hilliard fluids and topological transitions one of the fundamental problems in simulating the motion of sharp interfaces between immiscible fluids is a description of the transition that occurs when the interfaces merge and reconnect it is well known that classical methods involving sharp interfaces fail to describe this type of phenomena following some previous work in this area we suggest a physically motivated regularization of the euler equations which allows topological transitions to occur smoothly in this model the sharp interface is replaced by a narrow transition layer across which the fluids may mix the model describes a flow of a binary mixture and the internal structure of the interface is determined by both diffusion and motion an advantage of our regularization is that it automatically yields a continuous description of surface tension which can play an important role in topological transitions an additional scalar field is introduced to describe the concentration of one of the fluid components and the resulting system of equations couples the euler or navier stokes and the cahn hilliard equations the model takes into account weak non locality dispersion associated with an internal length scale and localized dissipation due to mixing the non locality introduces a dimensional surface energy dissipation is added to handle the loss of regularity of solutions to the sharp interface equations and to provide a mechanism for topological changes in particular we study a non trivial limit when both components are incompressible the pressure is kinematic but the velocity field is non solenoidal quasi incompressibility to demonstrate the effects of quasi incompressibility we analyse the linear stage of spinodal decomposition in one dimension we show that when the densities of the fluids are not perfectly matched the evolution of the concentration field causes fluid motion even if the fluids are inviscid in the limit of infinitely thin and well separated interfacial layers an appropriately scaled quasi incompressible euler cahn hilliard system converges to the classical sharp interface model in order to investigate the behaviour of the model outside the range of parameters where the sharp interface approximation is sufficient we consider a simple example of a change of topology and show that the model permits the transition to occur without an associated singularity

a phase field model for the mixture of two incompressible fluids and its approximation by a fourier spectral method abstract a phase field model for the mixture of two incompressible fluids is presented in this paper the model is based on an energetic variational formulation it consists of a navier stokes system linear momentum equation coupled with a cahn hilliard equation phase field equation through an extra stress term and the transport term the extra stress represents the phase induced capillary effect for the mixture due to the surface tension a fourier spectral method for the numerical approximation of this system is proposed and analyzed numerical results illustrating the robustness and versatility of the model are presented

finite element approximation of the nonstationary navier stokes problem i regularity of solutions and second order error estimates for spatial discretization this is the first part of a work dealing with the rigorous error analysis of finite element solutions of the nonstationary navier stokes equations second order error estimates are proven for spatial discretization using conforming or nonconforming elements the results indicate a fluid like behavior of the approximations even in the case of large data so long as the solution remains regular the analysis is based on sharp a priori estimates for the solution particularly reflecting its behavior as t to 0 and as t to infty it is shown that the regularity customarily assumed in the error analysis for corresponding parabolic problems cannot be realistically assumed in the case of the navier stokes equations as it depends on nonlocal compatibility conditions for the data the results which are presented here are independent of such compatibility conditions which cannot be verified in practice

efficient numerical solution of cahn hilliard navier stokes fluids in 2d a finite element discretization of the variable density cahn hilliard navier stokes system is presented this system is then decoupled with the cahn hilliard part solved via the nonlinear multigrid method proposed by the authors in j comput phys 212 2006 pp 288 304 and the navier stokes part via a multigrid preconditioned gmres scheme this new solver is examined numerically and is shown to be efficient with respect to mesh refinement and robust with respect to the problem parameters

convergence analysis and error estimates for a second order accurate finite element method for the cahn hilliard navier stokes system in this paper we present a novel second order in time mixed finite element scheme for the cahn hilliard navier stokes equations with matched densities the scheme combines a standard second order crank nicolson method for the navier stokes equations and a modification to the crank nicolson method for the cahn hilliard equation in particular a second order adams bashforth extrapolation and a trapezoidal rule are included to help preserve the energy stability natural to the cahn hilliard equation we show that our scheme is unconditionally energy stable with respect to a modification of the continuous free energy of the pde system specifically the discrete phase variable is shown to be bounded in ell infty left 0 t l infty right and the discrete chemical potential bounded in ell infty left 0 t l 2 right for any time and space step sizes in two and three dimensions and for any finite final time t we subsequently prove that these variables along with the fluid velocity converge with optimal rates in the appropriate energy norms in both two and three dimensions

on a diffuse interface model for two phase flows of viscous incompressible fluids with matched densities we study a diffuse interface model for the flow of two viscous incompressible newtonian fluids of the same density in a bounded domain the fluids are assumed to be macroscopically immiscible but a partial mixing in a small interfacial region is assumed in the model moreover diffusion of both components is taken into account this leads to a coupled navier stokes cahn hilliard system which is capable of describing the evolution of droplet formation and collision during the flow we prove the existence of weak solutions of the non stationary system in two and three space dimensions for a class of physical relevant and singular free energy densities which ensures in contrast to the usual case of a smooth free energy density that the concentration stays in the physical reasonable interval furthermore we find that unique strong solutions exist in two dimensions globally in time and in three dimensions locally in time moreover we show that for any weak solution the concentration is uniformly continuous in space and time because of this regularity we are able to show that any weak solution becomes regular for large times and converges as t to a solution of the stationary system these results are based on a regularity theory for the cahn hilliard equation with convection and singular potentials in spaces of fractional time regularity as well as on maximal regularity of a stokes system with variable viscosity and forces in l2 0 hs s in 0 frac12 which are new themselves

analysis of a darcy cahn hilliard diffuse interface model for the hele shaw flow and its fully discrete finite element approximation in this paper we present pde and finite element analyses for a system of pdes consisting of the darcy equation and the cahn hilliard equation which arises as a diffuse interface model for the two phase hele shaw flow in the model the two sets of equations are coupled through an extra phase induced force term in the darcy equations and a fluid induced transport term in the cahn hilliard equation we propose a fully discrete implicit finite element method for approximating the pde system which consists of the implicit euler method combined with a convex splitting energy strategy for the temporal discretization the standard finite element discretization for the pressure and a split or mixed finite element discretization for the fourth order cahn hilliard equation it is shown that the proposed numerical method satisfies a mass conservation law in addition to a discrete energy law that mimics the basic energy law for the darcy cahn hilliard phase field model and holds uniformly in the phase field

error analysis of a mixed finite element method for a cahn hilliard hele shaw system we present and analyze a mixed finite element numerical scheme for the cahn hilliard hele shaw equation a modified cahn hilliard equation coupled with the darcy flow law this numerical scheme was first reported in feng and wise siam j numer anal 50 1320 1343 2012 with the weak convergence to a weak solution proven in this article we provide an optimal rate error analysis a convex splitting approach is taken in the temporal discretization which in turn leads to the unique solvability and unconditional energy stability instead of the more standard ell infty 0 t l 2 cap ell 2 0 t h 2 l 0 t l2 l2 0 t h2 error estimate we perform a discrete ell infty 0 t h 1 cap ell 2 0 t h 3 l 0 t h1 l2 0 t h3 error estimate for the phase variable through an l 2 l2 inner product with the numerical error function associated with the chemical potential as a result an unconditional convergence for the time step tau in terms of the spatial resolution h is derived the nonlinear analysis is accomplished with the help of a discrete gagliardo nirenberg type inequality in the finite element space gotten by introducing a discrete laplacian delta h h of the numerical solution such that delta h phi in s h h sh for every phi in s h sh where s h sh is the finite element space

finite element approximations of the nonstationary navier stokes problem part iv error estimates for second order time discretization this paper provides an error analysis for the crank nicolson method of time discretization applied to spatially discrete galerkin approximations of the nonstationary navier stokes equations second order error estimates are proven locally in time under realistic assumptions about the regularity of the solution for approximations of an exponentially stable solution these local error estimates are extended uniformly in time as text t to infty

analysis of a mixed finite element method for a cahn hilliard darcy stokes system in this paper we devise and analyze a mixed finite element method for a modified cahn hilliard equation coupled with a nonsteady darcy stokes flow that models phase separation and coupled fluid flow in immiscible binary fluids and diblock copolymer melts the time discretization is based on a convex splitting of the energy of the equation we prove that our scheme is unconditionally energy stable with respect to a spatially discrete analogue of the continuous free energy of the system and unconditionally uniquely solvable we prove that the discrete phase variable is bounded in l infty left 0 t l infty right and the discrete chemical potential is bounded in l infty left 0 t l 2 right for any time and space step sizes in two and three dimensions and for any finite final time t we subsequently prove that these variables converge with optimal rates in the appropriate energy norms in both two and three dimensions

a second order in time uniquely solvable unconditionally stable numerical scheme for cahn hilliard navier stokes equation we propose a novel second order in time numerical scheme for cahn hilliard navier stokes phase field model with matched density the scheme is based on second order convex splitting for the cahn hilliard equation and pressure projection for the navier stokes equation we show that the scheme is mass conservative satisfies a modified energy law and is therefore unconditionally stable moreover we prove that the scheme is unconditionally uniquely solvable at each time step by exploring the monotonicity associated with the scheme thanks to the simple coupling of the scheme we design an efficient picard iteration procedure to further decouple the computation of cahn hilliard equation and navier stokes equation we implement the scheme by the mixed finite element method ample numerical experiments are performed to validate the accuracy and efficiency of the numerical scheme

fully discrete finite element approximations of the navier stokes cahn hilliard diffuse interface model for two phase fluid flows this paper develops and analyzes some fully discrete finite element methods for a parabolic system consisting of the navier stokes equations and the cahn hilliard equation which arises as a diffuse interface model for the flow of two immiscible and incompressible fluids in the model the two sets of equations are coupled through an extra phase induced stress term in the navier stokes equations and a fluid induced transport term in the cahn hilliard equation fully discrete mixed finite element methods are proposed for approximating the coupled system it is shown that the proposed numerical methods satisfy a mass conservation law and a discrete energy law which is analogous to the basic energy law for the phase field model the convergence of the numerical solutions to the solutions of the phase field model and its sharp interface limit is established by utilizing the discrete energy law as a by product the convergence result also provides a constructive proof of the existence of weak solutions to the navier stokes cahn hilliard phase field model numerical experiments are also presented to validate the theory and to show the effectiveness of the combined phase field and finite element approach

on convergent schemes for diffuse interface models for two phase flow of incompressible fluids with general mass densities we are concerned with convergence results for fully discrete finite element schemes suggested in grun and klingbeil j comput phys in press doi 10 1016 j jcp 2013 10 028 they were developed for the diffuse interface model in h abels h garcke and g grun math models methods appl sci 22 2012 1150013 which describes two phase flow of immiscible incompressible viscous fluids we formulate general conditions on discretization spaces and projection operators which allow us to prove compactness of discrete solutions with respect to both time and space and which hence permit us to establish convergence of the scheme to a generalized solution we identify a simple quantitative and physical criterion to decide whether this generalized solution is in fact a weak solution in this case our analysis provides another pathway to establish existence of weak solutions to the aforementioned model in two and in three space dimensions our argument is particularly based on higher regularity results

analysis of finite element approximations of a phase field model for two phase fluids this paper studies a phase field model for the mixture of two immiscible and incompressible fluids the model is described by a nonlinear parabolic system consisting of the nonstationary stokes equations coupled with the allen cahn equation through an extra phase induced stress term in the stokes equations and a fluid induced transport term in the allen cahn equation both semi discrete and fully discrete finite element methods are developed for approximating the parabolic system it is shown that the proposed numerical methods satisfy a discrete energy law which mimics the basic energy law for the phase field model error estimates are derived for the semi discrete method and the convergence to the phase field model and to its sharp interface limiting model are established for the fully discrete finite element method by making use of the discrete energy law numerical experiments are also presented to validate the theory and to show the effectiveness of the combined phase field and finite element approach

energy law preserving c0 finite element schemes for phase field models in two phase flow computations we use the idea in 33 to develop the energy law preserving method and compute the diffusive interface phase field models of allen cahn and cahn hilliard type respectively governing the motion of two phase incompressible flows we discretize these two models using a c 0 finite element in space and a modified midpoint scheme in time to increase the stability in the pressure variable we treat the divergence free condition by a penalty formulation under which the discrete energy law can still be derived for these diffusive interface models through an example we demonstrate that the energy law preserving method is beneficial for computing these multi phase flow models we also demonstrate that when applying the energy law preserving method to the model of cahn hilliard type un physical interfacial oscillations may occur we examine the source of such oscillations and a remedy is presented to eliminate the oscillations a few two phase incompressible flow examples are computed to show the good performance of our method

on the weil petersson metric on teichmuller space teichmuller space for a compact oriented surface m without boundary is described as the quotient ji b0 where s is the space of almost complex structures on m compatible with a given orientation and 8 0 are those c diffeomorphisms homotopic to the identity there is a natural 3 0 invariant l2 riemannian structure on si which induces a riemannian structure on si 3 n infinitesimally this is the bilinear pairing suggested by andre weil the weil petersson riemannian struc ture the structure is shown to be kahler with respect to a naturally induced complex structure on s0