mesoscopic simulation of ostwald ripening the self organization of particles in a two phase system in the coexistence region through a diffusive mechanism is known as ostwald ripening this phenomenon is an example of a multiscale problem in that the microscopic level interaction of the particles can greatly impact the macroscale or observable morphology of the system ostwald ripening is studied here through the use of a mesoscopic model which is a stochastic partial integrodifferential equation that is derived from a spin exchange ising model this model is studied through the use of recently developed and benchmarked spectral schemes for the simulation of solutions to stochastic partial differential equations the typical cluster size is observed to grow like t1 3 over range of times with faster growth at later times the results included here also demonstrate the effect of adjusting the interparticle interaction on the morphological evolution of the system at the macroscopic level

on the finite element method for singularly perturbed reaction diffusion problems in two and one dimensions second order elliptic boundary value problems which are allowed to degenerate into zero order equations are considered the behavior of the ordinary galerkin finite element method without special arrangements to treat singularities is studied as the problem ranges from true second order to singularly perturbed

an energy stable fourth order finite difference scheme for the cahn hilliard equation abstract in this paper we propose and analyze an energy stable numerical scheme for the cahn hilliard equation with second order accuracy in time and the fourth order finite difference approximation in space in particular the truncation error for the long stencil fourth order finite difference approximation over a uniform numerical grid with a periodic boundary condition is analyzed via the help of discrete fourier analysis instead of the standard taylor expansion this in turn results in a reduced regularity requirement for the test function in the temporal approximation we apply a second order bdf stencil combined with a second order extrapolation formula applied to the concave diffusion term as well as a second order artificial douglas dupont regularization term for the sake of energy stability as a result the unique solvability energy stability are established for the proposed numerical scheme and an optimal rate convergence analysis is derived in the l 0 t l 2 l 2 0 t h h 2 norm a few numerical experiments are presented which confirm the robustness and accuracy of the proposed scheme

a stable and conservative finite difference scheme for the cahn hilliard equation we propose a stable and conservative finite difference scheme to solve numerically the cahn hilliard equation which describes a phase separation phenomenon numerical solutions to the equation is hard to obtain because it is a nonlinear and nearly ill posed problem we design a new difference scheme based on a general strategy proposed recently by furihata and mori the new scheme inherits characteristic properties the conservation of mass and the decrease of the total energy from the equation the decrease of the total energy implies boundedness of discretized sobolev norm of the solution this in turn implies by discretized sobolev s lemma boundedness of max norm of the solution and hence the stability of the solution an error estimate for the solution is obtained and the order is o delta x 2 delta t 2 numerical examples demonstrate the effectiveness of the proposed scheme

finite element approximation of the cahn hilliard equation on surfaces abstract in this paper we consider the phase separation on general surfaces by solving the nonlinear cahn hilliard equation using a finite element method a fully discrete approximation scheme is introduced and we establish a priori estimates for the discrete solution that does not rely on any knowledge of the exact solution beyond the initial time this in turn leads to convergence and optimal error estimates of the discretization scheme numerical examples are also provided to substantiate the theoretical results

a second order splitting method for the cahn hilliard equation a semi discrete finite element method requiring only continuous element is presented for the approximation of the solution of the evolutionary fourth order in space cahn hilliard equation optimal order error bounds are derived in various norms for an implementation which uses mass lumping the continuous problem has an energy based lyapunov functional it is proved that this property holds for the discrete problem

nonconforming tetrahedral finite elements for fourth order elliptic equations this paper is devoted to the construction of nonconforming finite elements for the discretization of fourth order elliptic partial differential operators in three spatial dimensions the newly constructed elements include two nonconforming tetrahedral finite elements and one quasi conforming tetrahedral element these elements are proved to be convergent for a model bi harmonic equation in three dimensions in particular the quasi conforming tetrahedron element is a modified zienkiewicz element while the nonmodified zienkiewicz element a tetrahedral element of hermite type is proved to be divergent on a special grid

a second order accurate linearized difference scheme for the two dimensional cahn hilliard equation the cahn hilliard equation is a nonlinear evolutionary equation that is of fourth order in space in this paper a linearized finite difference scheme is derived by the method of reduction of order it is proved that the scheme is uniquely solvable and convergent with the convergence rate of order two in a discrete l2 norm the coefficient matrix of the difference system is symmetric and positive definite so many well known iterative methods e g gauss seidel sor can be used to solve the system

nonlinear aspects of the cahn hilliard equation abstract this paper treats phase separation within the context of the phenomenological cahn hilliard equation c t m c f c k 2 c where c is the concentration m c is the mobility and f c is homogeneou s free energy which is assumed here to be a fourth degree polynomial natural boundary conditions are introduced the full set of equilibrium solutions is specified a comparison theorem for stability criteria which was postulated by langer is proved here within the framework of the natural boundary conditions energy methods are used to define and estimate the limit of monotonic global stability it is pointed out that within the parameter region where the uniform homogeneous state is the only equilibrium solution there may still exist some internal excitable region in which the homogeneous solution possesses growing fluctuations furthermore a periodic instability is shown to exist in the metastable region in addition to the well known nucleation instability

a parallel domain decomposition based implicit method for the cahn hilliard cook phase field equation in 3d we present a numerical algorithm for simulating the spinodal decomposition described by the three dimensional cahn hilliard cook chc equation which is a fourth order stochastic partial differential equation with a noise term the equation is discretized in space and time based on a fully implicit cell centered finite difference scheme with an adaptive time stepping strategy designed to accelerate the progress to equilibrium at each time step a parallel newton krylov schwarz algorithm is used to solve the nonlinear system we discuss various numerical and computational challenges associated with the method the numerical scheme is validated by a comparison with an explicit scheme of high accuracy and unreasonably high cost we present steady state solutions of the chc equation in two and three dimensions the effect of the thermal fluctuation on the spinodal decomposition process is studied we show that the existence of the thermal fluctuation accelerates the spinodal decomposition process and that the final steady morphology is sensitive to the stochastic noise we also show the evolution of the energies and statistical moments in terms of the parallel performance it is found that the implicit domain decomposition approach scales well on supercomputers with a large number of processors

spectral method study of domain coarsening the self organization of particles in a two phase system in the coexistence region through a diffusive mechanism is known as ostwald ripening the late stage of ostwald ripening is studied here through the use of a mesoscopic model in conjunction with a special configuration that allows for the direct measurement of system characteristics such as domain size the mesoscopic model is a stochastic partial integrodifferential equation and is studied through the use of recently developed and benchmarked spectral schemes the size of the growing region is not observed to increase as a power law in this model during the late stages of self organization in contrast to the power law growth observed in simulations of the earlier stages of self organization the results included here also demonstrate the effect of adjusting the interparticle interaction on the morphological evolution of the system

the fourier collocation method for the cahn hilliard equation abstract in this paper a fourier collocation method for numerically solving cahn hilliard equations with periodic boundary conditions is developed we establish their semidiscrete and fully discrete schemes that inherit the energy dissipation property and mass conservation property from the associated continuous problem we prove existence and uniqueness of the numerical solution and derive the optimal error bounds we perform some numerical experiments which confirm our results

the fourier spectral method for the cahn hilliard equation in this paper a fourier spectral method for numerically solving cahn hilliard equation with periodic boundary conditions is developed we establish their semi discrete and fully discrete schemes that inherit the energy dissipation property and mass conservation property from the associated continuous problem we prove existence and uniqueness of the numerical solution and derive the optimal error bounds we perform some numerical experiments which confirm our results

analysis of a local discontinuous galerkin method for linear time dependent fourth order problems we analyze a local discontinuous galerkin method for fourth order time dependent problems optimal error estimates are obtained in one dimension and in multidimensions for cartesian and triangular meshes we extend the analysis to higher even order equations and the linearized cahn hilliard type equations numerical experiments are displayed to verify the theoretical results

simulation of the phase field cahn hilliard and tumor growth models via a numerical scheme element free galerkin method abstract the main aim of this research work is to find the numerical solution based on a meshless technique for both the time dependent cahn hilliard and tumor growth partial differential equations the temporal variable is discretized using a second order method based on semi implicit backward differential formula and the stabilized term is added to the considered time discretization also an adaptive time algorithm is used to reduce the number of iterations of the proposed time discretization besides to approximate the spatial variables the element free galerkin method is considered for both mathematical models the result of full discrete schemes in studied models is solved via biconjugate gradient stabilized algorithm some numerical simulations are reported to show the capability of the numerical scheme presented here

second order schemes and time step adaptivity for allen cahn and cahn hilliard models abstract in this paper we focus on efficient second order in time approximations of the allen cahn and cahn hilliard equations first of all we present the equations generic second order schemes based on a mid point approximation of the diffusion term and some schemes already introduced in the literature then we propose new ways of deriving second order in time approximations of the potential term starting from the main schemes introduced in guillen gonzalez and tierra 2013 yielding to new second order schemes for these schemes and other second order schemes previously introduced in the literature we study the constraints on the physical and discrete parameters that can appear to assure the energy stability unique solvability and in the case of nonlinear schemes the convergence of newton s method to the nonlinear schemes moreover in order to save computational cost we have developed a new adaptive time stepping algorithm based on the numerical dissipation introduced in the discrete energy law in each time step finally we compare the behaviour of the schemes and the effectiveness of the adaptive time stepping algorithm through several computational experiments

a c 1 virtual element method for the cahn hilliard equation with polygonal meshes in this paper we develop an evolution of the c 1 virtual elements of minimal degree for the approximation of the cahn hilliard equation the proposed method has the advantage of being conforming in h 2 and making use of a very simple set of degrees of freedom namely 3 degrees of freedom per vertex of the mesh moreover although the present method is new also on triangles it can make use of general polygonal meshes as a theoretical and practical support we prove the convergence of the semidiscrete scheme and investigate the performance of the fully discrete scheme through a set of numerical tests

a discontinuous galerkin method for stochastic cahn hilliard equations abstract in this paper a discontinuous galerkin method for the stochastic cahn hilliard equation with additive random noise which preserves the conservation of mass is investigated numerical analysis and error estimates are carried out for the linearized stochastic cahn hilliard equation the effects of the noises on the accuracy of our scheme are also presented numerical examples simulated by monte carlo method for both linear and nonlinear stochastic cahn hilliard equations are presented to illustrate the convergence rate and validate our conclusion

incidence of q statistics in rank distributions we show that size rank distributions with power law decay often only over a limited extent observed in a vast number of instances in a widespread family of systems obey tsallis statistics the theoretical framework for these distributions is analogous to that of a nonlinear iterated map near a tangent bifurcation for which the lyapunov exponent is negligible or vanishes the relevant statistical mechanical expressions associated with these distributions are derived from a maximum entropy principle with the use of two different constraints and the resulting duality of entropy indexes is seen to portray physically relevant information whereas the value of the index fixes the distribution s power law exponent that for the dual index 2 ensures the extensivity of the deformed entropy

approximation of smectic a liquid crystals in this paper we present a generic energy stable numerical scheme for a smectic a liquid crystal model this model involves the hydrodynamic velocity pressure variables u p and the order parameter of smectic a liquid crystals where its molecules have a uniaxial orientational order and a positional order by layers of normal given by the unitary vector n our starting point is the model presented by e in 6 using the so called layer variable such that n where is a strongly non linear parabolic system is considered coupling velocity and pressure of the navier stokes equations u p with a fourth order parabolic equation for we give a reformulation as a mixed second order problem which let us to define energy stable numerical schemes by using second order finite differences in time and c0 finite elements in space finally numerical simulations are presented for 2d domains showing the evolution of the system until it reaches an equilibrium configuration up to our knowledge there is not any previous numerical analysis for this model

error estimates with smooth and nonsmooth data for a finite element method for the cahn hilliard equation a finite element method for the cahn hilliard equation a semilinear parabolic equation of fourth order is analyzed both in a spatially semidiscrete case and in a completely discrete case based on the backward euler method error bounds of optimal order over a finite time interval are obtained for solutions with smooth and nonsmooth initial data a detailed study of the regularity of the exact solution is included the analysis is based on local lipschitz conditions for the nonlinearity with respect to sobolev norms and the existence of a ljapunov functional for the exact and the discretized equations is essential a result concerning the convergence of the attractor of the corresponding approximate nonlinear semigroup upper semicontinuity with respect to the discretization parameters is obtained as a simple application of the nonsmooth data error estimate

computationally efficient solution to the cahn hilliard equation adaptive implicit time schemes mesh sensitivity analysis and the 3d isoperimetric problem we present an efficient numerical framework for analyzing spinodal decomposition described by the cahn hilliard equation we focus on the analysis of various implicit time schemes for two and three dimensional problems we demonstrate that significant computational gains can be obtained by applying embedded higher order runge kutta methods in a time adaptive setting this allows accessing time scales that vary by five orders of magnitude in addition we also formulate a set of test problems that isolate each of the sub processes involved in spinodal decomposition interface creation and bulky phase coarsening we analyze the error fluctuations using these test problems on the split form of the cahn hilliard equation solved using the finite element method with basis functions of different orders any scheme that ensures at least four elements per interface satisfactorily captures both sub processes our findings show that linear basis functions have superior error to cost properties this strategy coupled with a domain decomposition based parallel implementation let us notably augment the efficiency of a numerical cahn hillard solver and open new venues for its practical applications especially when three dimensional problems are considered we use this framework to address the isoperimetric problem of identifying local solutions in the periodic cube in three dimensions the framework is able to generate all five hypothesized candidates for the local solution of periodic isoperimetric problem in 3d sphere cylinder lamella doubly periodic surface with genus two lawson surface and triply periodic minimal surface p schwarz surface

a nonconforming finite element method for the two dimensional cahn hilliard equation the cahn hilliard equation is a nonlinear evolutionary equation that is fourth order in space in this paper a continuous in time finite element galerkin approximation is considered we use the nonconforming morley element and derive optimal order error bounds in l 2

a class of stable spectral methods for the cahn hilliard equation in this work the initial boundary value problem of two dimensional cahn hilliard equation is considered a class of fully discrete dissipative fourier spectral schemes are proposed moreover semi implicit prediction correction schemes are presented the numerical simulations are performed to demonstrate the effectiveness of the proposed schemes

a fully discrete c0 interior penalty galerkin approximation of the extended fisher kolmogorov equation a fully discrete c 0 interior penalty finite element method is proposed and analyzed for the extended fisher kolmogorov efk equation u email protected d email protected u 3 u 0 with appropriate initial and boundary conditions where c is a positive constant we derive a regularity estimate for the solution u of the efk equation that is explicit in c and as a consequence we derive a priori error estimates that are robust in c

on linear schemes for a cahn hilliard diffuse interface model numerical schemes to approximate the cahn hilliard equation have been widely studied in recent times due to its connection with many physically motivated problems in this work we propose two type of linear schemes based on different ways to approximate the double well potential term the first idea developed in the paper allows us to design a linear numerical scheme which is optimal from the numerical dissipation point of view meanwhile the second one allows us to design unconditionally energy stable linear schemes for a modified energy we present first and second order in time linear schemes to approximate the ch problem detailing their advantages over other linear schemes that have been previously introduced in the literature furthermore we compare all the schemes through several computational experiments

new error estimates of the morley element for the plate bending problems in this paper we derive some new error estimates of a nonconforming morley element for the plate bending problems our aim is fourfold first it shows that the consistency error of the morley element can be controlled by its corresponding approximation error up to an arbitrarily high order term especially which can be of arbitrary order provided that f on the right hand side is piecewise smooth enough thus the total energy norm is dominated alone by the approximation error second as a byproduct we derive the error estimate under the regularity assumption u h 2 s with s 0 1 which fills the gap in the a priori error estimate of the morley element with low regularity for s 0 1 2 third based on this new error bound a robust convergence is proved even if the solution is exactly in h 2 finally by a slight modification of the variational formulation a new error estimate is also derived for the case f h 1 the key role is played by some quasi interpolation operators and the weak continuous properties of morley finite element space the main ideas and results presented here are motivated by mao and shi 2010 and can be viewed as its extension to fourth order problems

energy ssp imex runge kutta methods for the cahn hilliard equation we consider the cahn hilliard equation which describes phase separation phenomenon the equation is discretized by using a fourth order compact difference scheme in space and strong stability preserving ssp implicit explicit imex runge kutta methods in time the new methods have two distinct features 1 the large time steps can be used in the numerical simulation because the energy is stable and 2 the energy functional decreases by time unconditional energy stability of first order and second order methods are proved numerical experiments are given to demonstrate the performance of the proposed methods

stable finite difference scheme for a model equation of phase separation a stable conservative and practical finite difference scheme is presented in order to solve numerically a model equation of phase separation in some binary alloys the model system is introduced by eguchi oki and matsumura in the attempt of theoretically describing such phenomena and extends the well known cahn hilliard equation even numerical solutions are hard to obtain for this dynamics our scheme utilizes a lyapunov functional that is the free energy of the system whose effectiveness is demonstrated by numerical implementations

a time adaptive finite volume method for the cahn hilliard and kuramoto sivashinsky equations this paper presents a complete finite volume method for the cahn hilliard and kuramoto sivashinsky type of equations the spatial discretization is high order accurate and suitable for general unstructured grids the time integration is addressed by means of implicit an implicit explicit fourth order runge kutta schemes with error control and adaptive time stepping the outcome is a practical accurate and efficient simulation tool which has been successfully applied to accuracy tests and representative simulations the use of adaptive time stepping is of paramount importance in problems governed by the cahn hilliard model an adaptive method may be several orders of magnitude more efficient than schemes using constant or heuristic time steps in addition to driving the simulations efficiently the time adaptive procedure provides a quantitative not just qualitative characterization of the rich temporal scales present in phase separation processes governed by the cahn hilliard phase field model

a second order accurate non linear difference scheme for the n component cahn hilliard system a b s t r a c t we consider a second order conservative nonlinear numerical scheme for the ncomponent cahn hilliard system modeling the phase separation of a n component mixture the scheme is based on a crank nicolson finite difference method and is solved by an efficient and accurate nonlinear multigrid method we numerically demonstrate the second order accuracy of the numerical scheme we observe that our numerical solutions are consistent with the exact solutions of linear stability analysis results we also describe numerical experiments such as the evolution of triple junctions and the spinodal decomposition in a quaternary mixture we investigate the effects of a concentration dependent mobility on phase separation

error analysis of a mixed finite element method for the cahn hilliard equation we propose and analyze a semi discrete and a fully discrete mixed finite element method for the cahn hilliard equation ut u 1f u 0 where 0 is a small parameter error estimates which are quasi optimal order in time and optimal order in space are shown for the proposed methods under minimum regularity assumptions on the initial data and the domain in particular it is shown that all error bounds depend on only in some lower polynomial order for small the cruxes of our analysis are to establish stability estimates for the discrete solutions to use a spectrum estimate result of alikakos and fusco 2 and chen 15 to prove a discrete counterpart of it for a linearized cahn hilliard operator to handle the nonlinear term on a stretched time grid the ideas and techniques developed in this paper also enable us to prove convergence of the fully discrete finite element solution to the solution of the hele shaw mullins sekerka problem as 0 in 29

a discontinuous galerkin method for the cahn hilliard equation a discontinuous galerkin finite element method has been developed to treat the high order spatial derivatives appearing in the cahn hilliard equation the cahn hilliard equation is a fourth order nonlinear parabolic partial differential equation originally proposed to model phase segregation of binary alloys the developed discontinuous galerkin approach avoids the need for mixed finite element methods coupled equations or interpolation functions with a high degree of continuity that have been employed in the literature to treat the fourth order spatial derivatives the variational formulation of the discontinuous galerkin method its implementation and numerical examples are presented in this communication it is also shown under what conditions the method is stable and an error estimate in an energy type norm is presented the method is evaluated by comparison with a standard finite element treatment in which the cahn hilliard equation is decomposed into two coupled partial differential equations

local discontinuous galerkin methods for the cahn hilliard type equations in this paper we develop local discontinuous galerkin ldg methods for the fourth order nonlinear cahn hilliard equation and system the energy stability of the ldg methods is proved for the general nonlinear case numerical examples for the cahn hilliard equation and the cahn hilliard system in one and two dimensions are presented and the numerical results illustrate the accuracy and capability of the methods

analysis of mixed interior penalty discontinuous galerkin methods for the cahn hilliard equation and the hele shaw flow this paper proposes and analyzes two fully discrete mixed interior penalty discontinuous galerkin dg methods for the fourth order nonlinear cahn hilliard equation both methods use the backward euler method for time discretization and interior penalty dg methods for spatial discretization they differ from each other on how the nonlinear term is treated one of them is based on fully implicit time stepping and the other uses energy splitting time stepping the primary goal of the paper is to prove the convergence of the numerical interfaces of the dg methods to the interface of the hele shaw flow this is achieved by establishing error estimates that depend on epsilon 1 only in some low polynomial orders instead of exponential orders similar to x feng and a prohl numer math 74 2004 pp 47 84 the crux is to prove a discrete spectrum estimate in the discontinuous galerkin finite element space however the validity of such a result is not obvious because the dg space is not a subspa