shallow deep and very deep cuts in the analytic center cutting plane method in this paper we show that the cut does not need to go through the query point it can be deep or shallow the primal framework leads to a simple analysis of the potential variation which shows that the inequality needed for convergence of the algorithm is in fact attained at the first iterate of the feasibility step

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

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

finite element approximation of the cahn hilliard equation with degenerate mobility we consider a fully practical finite element approximation of the cahn hilliard equation with degenerate mobility textstyle frac partial u partial t del b u del gamma lap u psi u where b cdot geq 0 is a diffusional mobility and psi cdot is a homogeneous free energy in addition to showing well posedness and stability bounds for our approximation we prove convergence in one space dimension furthermore an iterative scheme for solving the resulting nonlinear discrete system is analyzed we also discuss how our approximation has to be modified in order to be applicable to a logarithmic homogeneous free energy finally some numerical experiments are presented

a multiple cut analytic center cutting plane method for semidefinite feasibility problems we consider the problem of finding a point in a nonempty bounded convex body gamma in the cone of symmetric positive semidefinite matrices cal s m assume that gamma is defined by a separating oracle which for any given m ti m symmetric matrix hat y either confirms that hat y in gamma or returns several selected cuts i e a number of symmetric matrices ai i 1 p p le p max such that gamma is in the polyhedron y in cal s m mid a i bullet y le a i bullet hat y i 1 ldots p we present a multiple cut analytic center cutting plane algorithm starting from a trivial initial point the algorithm generates a sequence of positive definite matrices which are approximate analytic centers of a shrinking polytope in cal s m the algorithm terminates with a point in gamma within o m 3p max epsilon 2 newton steps to leading order where epsilon is the maximum radius of a ball contained in gamma

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

stability analysis and error estimates of local discontinuous galerkin methods with semi implicit spectral deferred correction time marching for the allen cahn equation abstract this paper is concerned with the stability and error estimates of the local discontinuous galerkin ldg method coupled with semi implicit spectral deferred correction sdc time marching up to third order accuracy for the allen cahn equation since the sdc method is based on the first order convex splitting scheme the implicit treatment of the nonlinear item results in a nonlinear system of equations at each step which increases the difficulty of the theoretical analysis for the ldg discretizations coupled with the second and third order sdc methods we prove the unique solvability of the numerical solutions through the standard fixed point argument in finite dimensional spaces at the same time the iteration and integral involved in the semi implicit sdc scheme also increase the difficulty of the theoretical analysis comparing to the runge kutta type semi implicit schemes which exclude the left most endpoint the sdc scheme in this paper includes the left most endpoint as a quadrature node this makes the test functions of the sdc scheme more complicated and the energy equations are more difficult to construct we provide two different ideas to overcome the difficulty of the nonlinear terms by choosing the test functions carefully the energy stability and error estimates are obtained in the sense that the time step t only requires a positive upper bound and is independent of the mesh size h numerical examples are presented to illustrate our theoretical results

systems of cahn hilliard equations the phase separation of alloys with two or more components is studied with emphasis on more than two components particular attention is given to differences between multicomponent and binary alloys specific topics of the paper include equilibrium theory aspects of the dynamics and numerical simulations in the equilibrium theory it is found that there is an enriched equilibrium structure that allows for multiple coexisting phases and the presence of triple points in the solution dynamic results include the characterization of the spinodal region and of the concentration variations that lead to spinodal decomposition unlike the binary theory not all composition variations lead to separation and the compositions are not restricted to the convex hull of the equilibrium concentrations linear analysis is used to predict that a pseudo binary will initially result from spinodal decomposition numerical simulations of the dynamics for a ternary alloy verify this initially but more than two phases often

error estimate procedure in the finite element method and applications statically and kinematically admissible fields are explicitly derived from the finite element solution of the primal form of linear models the error on constitutive law for these fields yields an expression of the finite element error moreover the contribution of each element to this error allows to implement an automatic mesh refinement procedure leading to a uniform distribution of a given accuracy

on vaidya s volumetric cutting plane method for convex programming we describe a simplified and strengthened version of vaidya s volumetric cutting plane method for finding a point in a convex set x1d49e rn at each step the algorithm has a system of linear inequality constraints which defines a polyhedron x1d4ab x1d49e and an interior point x x1d4ab the algorithm then either drops one constraint or calls an oracle to check if x x1d49e and if not obtain a new constraint that separates x from x1d49e following the addition or deletion of a constraint the algorithm takes a small number of newton steps for the volumetric barrier v progress of the algorithm is measured in terms of changes in v the algorithm is terminated when either it is discovered that x x1d49e or v becomes large enough to demonstrate that the volume of x1d49e must be below some prescribed amount the complexity of the algorithm compares favorably with that of the ellipsoid method especially in terms of the number of calls to the separation oracle compared to vaidya s original analysis we decrease the total number

complexity analysis of the analytic center cutting plane method that uses multiple cuts we analyze the complexity of the analytic center cutting plane or column generation algorithm for solving general convex problems defined by a separation oracle the oracle is called at the analytic center of a polytope which contains a solution set and is given by the intersection of the linear inequalities previously generated from the oracle if the center is not in the solution set separating hyperplanes will be placed through the center to shrink the containing polytope while the complexity result has been recently established for the algorithm when one cutting plane is placed in each iteration the result remains open when multiple cuts are added moreover adding multiple cuts actually is a key to practical effectiveness in solving many problems and it presents theoretical difficulties in analyzing cutting plane methods in this paper we show that the analytic center cutting plane algorithm with multiple cuts added in each iteration still is a fully polynomial approximation algorithm

towards a practical volumetric cutting plane method for convex programming we consider the volumetric cutting plane method for finding a point in a convex set cal c subset re n that is characterized by a separation oracle we prove polynomiality of the algorithm with each added cut placed directly through the current point and show that this central cut version of the method can be implemented using no more than 25n constraints at any time

a comment on the article a finite element method for simulation of strong and weak discontinuities in solid mechanics by a hansbo and p hansbo comput methods appl mech engrg 193 2004 3523 3540

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

an efficient unconditionally energy stable local discontinuous galerkin scheme for the cahn hilliard brinkman system in this paper we present an efficient and unconditionally energy stable fully discrete local discontinuous galerkin ldg method for approximating the cahn hilliard brinkman chb system which is comprised of a cahn hilliard type equation and a generalized brinkman equation modeling fluid flow the semi discrete energy stability of the ldg method is proved firstly due to the strict time step restriction t o x 4 of explicit time discretization methods for stability we introduce a semi implicit scheme which consists of the implicit euler method combined with a convex splitting of the discrete cahn hilliard energy strategy for the temporal discretization the unconditional energy stability of this fully discrete convex splitting scheme is also proved obviously the fully discrete equations at the implicit time level are nonlinear and to enhance the efficiency of the proposed approach the nonlinear full approximation scheme fas multigrid method has been employed to solve this system of algebraic equations we also show the nearly optimal complexity numerically numerical experiments based on the overall solution method of combining the proposed ldg method convex splitting scheme and the nonlinear multigrid solver are given to validate the theoretical results and to show the effectiveness of the proposed approach for the chb system

an analytic center based column generation algorithm for convex quadratic feasibility problems we consider the analytic center based column generation algorithm for the problem of finding a feasible point in a set defined by a finite number of convex quadratic inequalities at each iteration the algorithm computes an approximate analytic center of the set defined by the intersection of quadratic inequalities generated in the previous iterations if this approximate analytic center is a solution then the algorithm terminates otherwise a quadratic inequality violated at the current center is selected and a new quadratic cut defined by a convex quadratic inequality is placed near the approximate center as the number of cuts increases the set defined by their intersection shrinks and the algorithm eventually finds a solution to the problem previously similar analytic center based column generation algorithms were studied first for the linear feasibility problem and later for the general convex feasibility problem our method differs from these early methods in that we use quadratic cuts in the computation instead of linear cuts moreover our method has a polynomial worst case complexity of o n ln frac 1 varepsilon on the total number of cuts to be used where n is the number of convex quadratic polynomial inequalities in the problem and varepsilon is the radius of the largest ball contained in the feasible set in contrast the early column generation methods using linear cuts can only solve the convex quadratic feasibility problem in pseudopolynomial time

extended finite element method for three dimensional crack modelling an extended finite element method x fem for three dimensional crack modelling is described a discontinuous function and the two dimensional asymptotic crack tip displacement fields are added to the finite element approximation to account for the crack using the notion of partition of unity this enables the domain to be modelled by finite elements with no explicit meshing of the crack surfaces computational geometry issues associated with the representation of the crack and the enrichment of the finite element approximation are discussed stress intensity factors sifs for planar three dimensional cracks are presented which are found to be in good agreement with benchmark solutions copyright 2000 john wiley sons ltd

finite element approximation of a model for phase separation of a multi component alloy with non smooth free energy a model for the phase separation of a multi component alloy with non smooth free energy is considered an error bound is proved for a fully practical piecewise linear finite element approximation using a backward euler time discretization an iterative scheme for solving the resulting nonlinear algebraic system is analysed finally numerical experiments with three components in one and two space dimensions are presented in the one dimensional case we compare some steady states obtained numerically with the corresponding stationary solutions of the continuous problem which we construct explicitly

analysis of a novel finite element method for a modified cahn hilliard hele shaw system abstract in this paper a novel finite element method for solving a modified cahn hilliard hele shaw system is proposed the time discretization is based on the convex splitting of the energy functional in the modified cahn hilliard equation i e the high order nonlinear term and the linear term in the chemical potential are treated explicitly and implicitly respectively designing in this way leads to solving a linear system at each time step which is much efficient compared to solving a nonlinear system resulting from most existing schemes the proposed scheme is proved to be unconditionally energy stable and optimally convergent for the phase variable numerical results are presented to support our theoretical analysis

unconditionally stable methods for simulating multi component two phase interface models with peng robinson equation of state and various boundary conditions in this paper we consider multi component dynamic two phase interface models which are formulated by the cahn hilliard system with peng robinson equation of state and various boundary conditions these models can be derived from the minimum problems of helmholtz free energy or grand potential in the realistic thermodynamic systems the resulted cahn hilliard systems with various boundary conditions are fully coupled and strongly nonlinear a linear transformation is introduced to decouple the relations between different components and as a result the models are simplified from this we further propose a semi implicit unconditionally stable time discretization scheme which allows us to solve the cahn hilliard system by a decoupled way and thus our method can significantly reduce the computational cost and memory requirements the mixed finite element methods are employed for the spatial discretization and the approximate errors are also analyzed for both space and time numerical examples are tested to demonstrate the efficiency of our proposed methods

analysis of a new finite difference local discontinuous galerkin method for the fractional diffusion wave equation in this paper a finite difference local discontinuous galerkin method for the fractional diffusion wave equation is presented and analyzed we first propose a new finite difference method to approximate the time fractional derivatives and give a semidiscrete scheme in time further we develop a fully discrete scheme for the fractional diffusion wave equation and prove that the method is unconditionally stable and convergent with order o hk 1 t 3 where k is the degree of piecewise polynomial extensive numerical examples are carried out to confirm the theoretical convergence rates

an efficient fully discrete local discontinuous galerkin method for the cahn hilliard hele shaw system in this paper we develop an efficient and energy stable fully discrete local discontinuous galerkin ldg method for the cahn hilliard hele shaw chhs system the semi discrete energy stability of the ldg method is proved firstly due to the strict time step restriction dt o dx 4 of explicit time discretization methods for stability we introduce a semi implicit time integration scheme which is based on a convex splitting of the discrete cahn hilliard energy the unconditional energy stability has been proved for this fully discrete ldg scheme the fully discrete equations at the implicit time level are nonlinear thus the nonlinear full approximation scheme fas multigrid method has been applied to solve this system of algebraic equations which has been shown the nearly optimal complexity numerically numerical results are also given to illustrate the accuracy and capability of the ldg method coupled with the multigrid solver

a multigrid finite element solver for the cahn hilliard equation a multigrid finite element solver for the cahn hilliard equation is presented that has mesh independent convergence rates for any time step size including in the important limit e 0 which is examined via numerical examples numerics are performed for a number of test problems which show that the features of the cahn hilliard equation minimising interface measure lyapunov energy functional etc are preserved we also explore the use of this solver in conjunction with adaptive time stepping and adaptive mesh strategies

derivative recovery and a posteriori error estimate for extended finite elements this paper is the first attempt at error estimation for extended finite elements the goal of this work is to devise a simple and effective local a posteriori error estimate for partition of unity enriched finite element methods such as the extended finite element method xfem in each element the local estimator is the l2 norm of the difference between the raw xfem strain field and an enhanced strain field computed by extended moving least squares xmls derivative recovery obtained from the raw nodal xfem displacements the xmls construction is tailored to the nature of the solution the technique is applied to linear elastic fracture mechanics in which near tip asymptotic functions are added to the mls basis the xmls shape functions are constructed from weight functions following the diffraction criterion to represent the discontinuity the result is a very smooth enhanced strain solution including the singularity at the crack tip results are shown for two and three dimensional linear elastic fracture mechanics problems in mode i and mixed mode the effectivity index of the estimator is close to 1 and improves upon mesh refinement for the studied near tip problem it is also shown that for the linear elastic fracture mechanics problems treated the proposed estimator outperforms one of the superconvergent patch recovery technique of zienkiewicz and zhu which is only c0 parametric studies of the general performance of the estimator are also carried out

the superconvergent patch recovery and a posteriori error estimates part 2 error estimates and adaptivity in this second part of the paper the issue of a posteriori error estimation is discussed in particular we derive a theorem showing the dependence of the effectivity index for the zienkiewicz zhu error estimator on the convergence rate of the recovered solution this shows that with superconvergent recovery the effectivity index tends asymptotically to unity the superconvergent recovery technique developed in the first part of the paper1 is the used in the computation of the zienkiewicz zhu error estimator to demonstrate accurate estimation of the exact error attainable numerical tests are shown for various element types illustrating the excellent effectivity of the error estimator in the energy norm and pointwise gradient stress error estimation several examples of the performance of the error estimator in adaptive mesh refinement are also presented

analysis of a cutting plane method that uses weighted analytic center and multiple cuts we consider the analytic center cutting plane or column generation algorithm for solving general convex problems defined by a separation oracle the oracle is called at an approximate analytic center of a polytope which contains the solution set and is given by the intersection of the linear inequalities previously generated from the oracle if the approximate center is not in the solution set separating hyperplanes will be placed through the approximate center and a new approximate analytic center will be found for the shrunken polytope in this paper we consider using approximate weighted analytic centers in the cutting plane method and show that the method with multiple cuts added in each step has a complexity of o eta m 2 epsilon 2 where eta is the maximum number of cuts that can be added in each step and m is the dimension of the problem

a fully discrete stable discontinuous galerkin method for the thin film epitaxy problem without slope selection in this paper we develop an energy stable fully discrete discontinuous galerkin dg finite element method for the thin film epitaxy problem based on the method of lines we construct and prove the energy stability of the spatial semi discrete dg scheme firstly to avoid the strict time step restriction of the explicit time integration method the first order convex splitting method is used to get an unconditionally stable fully discrete dg method which is a linearly implicit scheme for this nonlinear problem the energy stability of the fully discrete convex splitting dg scheme is also proved to improve the temporal accuracy spectral deferred correction sdc method is adapted to achieve the high order accuracy in both time and space combining with the convex splitting method the sdc method can be linearly solvable high order accurate and stable in our numerical tests these advantages ensure that the resulting fully discrete dg scheme is efficient to perform the long time simulation of the thin film epitaxy model numerical experiments of the accuracy and long time simulation show the capability and efficiency of the method