local discontinuous galerkin methods for nonlinear schrodinger equations in this paper we develop a local discontinuous galerkin method to solve the generalized nonlinear schrodinger equation and the coupled nonlinear schrodinger equation l 2 stability of the schemes are obtained for both of these nonlinear equations numerical examples are shown to demonstrate the accuracy and capability of these methods

numerical studies of discrete approximations to the allen cahn equation in the sharp interface limit the numerical approximations to the allen cahn type diffuse interface models are studied with a particular focus on their performance in the sharp interface limit and the effectiveness of high order discretization schemes different spatial discretizations of an energy functional in the diffuse interface framework are compared first discretizations of the time dependent equation using various different time stepping schemes and an adaptive finite element spatial approximations are then analyzed assessments of the numerical accuracy in different parameter regimes are carried out the analysis and computational findings provide insight on the effectiveness of the discretization schemes and offer guidance to parameter choices when numerical simulations are conducted within the diffuse interface framework

local discontinuous galerkin method and high order semi implicit scheme for the phase field crystal equation in this paper we present a local discontinuous galerkin ldg method and two unconditionally energy stable schemes for the phase field crystal pfc equation the semidiscrete energy stability of the ldg method is proved first the pfc equation is a sixth order nonlinear partial differential equation pde which leads to the severe time step restriction delta t mathcal o delta x 6 of explicit time discretization methods to maintain stability due to this we introduce semi implicit first order and second order time discretization methods which are based on the convex splitting principle of a discrete energy and prove the corresponding unconditional energy stabilities to improve the temporal accuracy the spectral deferred correction sdc method and a high order semi implicit runge kutta method combining with the first order convex splitting method are adopted for the pfc equation with constant and degenerate mobility respectively the equations at the implicit time level are nonlinear and

stability and error estimates of local discontinuous galerkin methods with implicit explicit time marching for advection diffusion problems the main purpose of this paper is to analyze the stability and error estimates of the local discontinuous galerkin ldg methods coupled with carefully chosen implicit explicit imex runge kutta time discretization up to third order accuracy for solving one dimensional linear advection diffusion equations in the time discretization the advection term is treated explicitly and the diffusion term implicitly there are three highlights of this work the first is that we establish an important relationship between the gradient and interface jump of the numerical solution with the independent numerical solution of the gradient in the ldg methods the second is that by aid of the aforementioned relationship and the energy method we show that the imex ldg schemes are unconditionally stable for the linear problems in the sense that the time step tau is only required to be upper bounded by a constant which depends on the ratio of the diffusion and the square of the advection coefficients and is independent

discontinuous galerkin finite element approximation of the cahn hilliard equation with convection the paper is concerned with the construction and convergence analysis of a discontinuous galerkin finite element method for the cahn hilliard equation with convection using discontinuous piecewise polynomials of degree p geq1 and backward euler discretization in time we show that the order parameter c is approximated in the broken rm l infty rm h 1 norm with optimal order cal o h p tau the associated chemical potential w phi c gamma 2 delta c is shown to be approximated with optimal order cal o h p tau in the broken rm l 2 rm h 1 norm here phi c frac 1 4 1 c 2 2 is a quartic free energy function and gamma 0 is an interface parameter numerical results are presented with polynomials of degree p 1 2 3

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

discontinuous galerkin methods for the helmholtz equation with large wave number this paper develops and analyzes some interior penalty discontinuous galerkin ipdg methods using piecewise linear polynomials for the helmholtz equation with the first order absorbing boundary condition in two and three dimensions it is proved that the proposed discontinuous galerkin methods are stable hence well posed without any mesh constraint for each fixed wave number k optimal order with respect to h error estimate in the broken h 1 norm and suboptimal order estimate in the l 2 norm are derived without any mesh constraint the latter estimate improves to optimal order when the mesh size h is restricted to the preasymptotic regime i e k 2 h gtrsim 1 numerical experiments are also presented to gauge the theoretical result and to numerically examine the pollution effect with respect to k in the error bounds the novelties of the proposed ipdg methods include the following first the methods penalize not only the jumps of the function values across the element edges but also the jumps of the normal and tangential derivatives second the penalty parameters are taken as complex numbers of positive imaginary parts so essentially and practically no constraint is imposed on the penalty parameters since the helmholtz problem is a non hermitian and indefinite linear problem as expected the crucial and the most difficult part of the whole analysis is to establish the stability estimates i e a priori estimates for the numerical solutions to this end the cruxes of our analysis are to establish and to make use of a local version of the rellich identity for the laplacian and to mimic the stability analysis for the pde solutions given in p cummings analysis of finite element based numerical methods for acoustic waves elastic waves and fluid solid interactions in the frequency domain ph d thesis the university of tennessee knoxville tn 2001 p cummings and x feng math models methods appl sci 16 2006 pp 139 160 u hetmaniuk commun math sci 5 2007 pp 665 678

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

coherent states and the role of the affine group in the quantum mechanics of the morse potential the coherent states of the morse potential that have been obtained earlier from supersymmetric quantum mechanics are shown to be connected with the representations of the affine group of the real line and some of its extensions this relation is similar to the one between the heisenberg weyl group and the coherent states of the harmonic oscillator the states that minimize the uncertainty product of the generators of the affine lie algebra are shown to contain all the coherent states of the morse oscillator plus the intelligent states of the morse hamiltonians with different shape parameter s the representations of the central extension of the affine group denoted by g0 and its further extension will be shown to define the phase space relevant to the problem by choosing an appropriate orbit of the coadjoint representation of this allows one to construct a generalized wigner function on this phase space which is again essentially in the same relation with the affine group as the ordinary wigner function with the heisenberg weyl group

stability analysis and error estimates of local discontinuous galerkin methods with implicit explicit time marching for nonlinear convection diffusion problems the main purpose of this paper is to analyze the stability and error estimates of the local discontinuous galerkin ldg methods coupled with implicit explicit imex time discretization schemes for solving one dimensional convection diffusion equations with a nonlinear convection both runge kutta and multi step imex methods are considered by the aid of the energy method we show that the imex ldg schemes are unconditionally stable for the nonlinear problems in the sense that the time step is only required to be upper bounded by a positive constant which depends on the flow velocity and the diffusion coefficient but is independent of the mesh size h we also give optimal error estimates for the imex ldg schemes under the same temporal condition if a monotone numerical flux is adopted for the convection numerical experiments are given to verify our main results

piecewise solenoidal vector fields and the stokes problem nonconforming finite element approximations to solutions of the stokes equations are constructed optimal rates of convergence are proved for the velocity and pressure approximations for the pressure approximation c 0 piecewise polynomial functions are used the class of vector fields used to approximate the velocity field have piecewise polynomial components discontinuous across interelement boundaries on each triangle these vector fields satisfy the incompressibility condition pointwise it is shown that these piecewise solenoidal vector fields possess optimal approximation properties to smooth solenoidal vector fields on domains with curved boundaries

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

milnor moore categories and monadic decomposition abstract in this paper monoidal hom lie algebras lie color algebras lie superalgebras and other type of generalized lie algebras are recovered by means of an iterated construction known as monadic decomposition of functors which is based on eilenberg moore categories to this aim we introduce the notion of milnor moore category as a monoidal category for which a milnor moore type theorem holds we also show how to lift the property of being a milnor moore category whenever a suitable monoidal functor is given and we apply this technique to provide examples

infiniteness of zero modes for the pauli operator with singular magnetic field abstract we establish that the pauli operator describing a spin 1 2 two dimensional quantum system with a singular magnetic field has under certain conditions an infinite dimensional space of zero modes possibly both spin up and spin down moreover there is a spectral gap separating the zero eigenvalue from the rest of the spectrum in particular infiniteness takes place if the field has infinite flux which settles this previously unknown case of aharonov casher theorem

convergence of the cahn hilliard equation to the hele shaw model we prove that level surfaces of solutions to the cahn hilliard equation tend to solutions of the hele shaw problem under the assumption that classical solutions of the latter exist the method is based on a new matched asymptotic expansion for solutions a spectral analysis for linearizd operators and an estimate for the difference between the true solutions and certain approximate ones

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

essential self adjointness for semi bounded magnetic schrodinger operators on non compact manifolds abstract we prove essential self adjointness for semi bounded below magnetic schrodinger operators on complete riemannian manifolds with a given positive smooth measure which is fixed independently of the metric some singularities of the scalar potential are allowed this is an extension of the povzner wienholtz simader theorem the proof uses the scheme of wienholtz but requires a refined invariant integration by parts technique as well as the use of a family of cut off functions which are constructed by a non trivial smoothing procedure due to karcher

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

pre asymptotic error analysis of h p interior penalty discontinuous galerkin methods for the helmholtz equation with large wave number in this paper we shall consider to improve the pre asymptotic stability and error estimates of some h p interior penalty discontinuous galerkin h p ipdg methods for the helmholtz equation with the first order absorbing boundary condition in two and three dimensions given in feng and wu 2011 the proposed h p ipdg methods are defined using a sesquilinear form which is not only mesh dependent or h dependent but also degree dependent or p dependent by using a modified duality argument given in zhu and wu 2013 pre asymptotic error estimates are improved for the proposed h p ipdg methods under the condition of k h p c 0 p k 1 p 1 in this paper where c 0 is some constant independent of k h p and the penalty parameters it is shown that the pollution error of the method in the broken h 1 norm is o k 2 p 1 h 2 p if p o 1 which coincides with existent dispersion analyses for the dg method on cartesian grids numerical tests are provided to verify the theoretical findings and to illustrate great capability of the ipdg method in reducing the pollution effect

discontinuous finite element methods for a bi wave equation modeling d wave superconductors this paper concerns discontinuous nite element approximations of a fourth order bi wave equation arising as a simplied ginzburg landau type model for d wave superconductors in the absence of an applied magnetic eld in the rst half of the paper we construct a variant of the morley nite element method which was originally developed for approximating the fourth order biharmonic equation for the bi wave equation it is proved that unlike the biharmonic equation it is necessary to impose a mesh constraint and to include certain penalty terms in the method to guarantee convergence nearly optimal order o by a factor j lnhj error estimates in the energy norm and in the h 1 norm are established for the proposed morley type nonconforming method in the second half of the paper we develop a symmetric interior penalty discontinuous galerkin method for the bi wave equation using general meshes and prove optimal order error estimates in the energy norm finally numerical experiments are provided to gauge the eciency of the proposed methods and to validate the theoretical error bounds

discontinuous hp finite element methods for advection diffusion reaction problems we consider the hp version of the discontinuous galerkin finite element method dgfem for second order partial differential equations with nonnegative characteristic form this class of equations includes second order elliptic and parabolic equations advection reaction equations as well as problems of mixed hyperbolic elliptic parabolic type our main concern is the error analysis of the method in the absence of streamline diffusion stabilization in the hyperbolic case an hp optimal error bound is derived here we consider only advection reaction problems which satisfy a certain standard positivity condition in the self adjoint elliptic case an error bound that is h optimal and p suboptimal by frac 1 2 a power of p is obtained these estimates are then combined to deduce an error bound in the general case for elementwise analytic solutions the method exhibits exponential rates of convergence under p refinement the theoretical results are illustrated by numerical experiments

optimal error estimates of the semidiscrete local discontinuous galerkin methods for high order wave equations in this paper we introduce a general approach for proving optimal l 2 error estimates for the semidiscrete local discontinuous galerkin ldg methods solving linear high order wave equations the optimal order of error estimates holds not only for the solution itself but also for the auxiliary variables in the ldg method approximating the various order derivatives of the solution examples including the one dimensional third order wave equation one dimensional fifth order wave equation and multidimensional schrodinger equation are explored to demonstrate this approach the main idea is to derive energy stability for the various auxiliary variables in the ldg discretization by using the scheme and its time derivatives with different test functions special projections are utilized to eliminate the jump terms at the cell boundaries in the error estimate in order to achieve the optimal order of accuracy

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

aharonov bohm effect with δ type interaction a quantum particle interacting with a thin solenoid and a magnetic flux is described by a five parameter family of hamilton operators obtained via the method of self adjoint extensions one of the parameters the value of the flux corresponds to the aharonov bohm effect the other four parameters correspond to the strength of a singular potential barrier the spectrum and eigenstates are computed and the scattering problem is solved

a nonconforming finite element method for the cahn hilliard equation this paper reports a fully discretized scheme for the cahn hilliard equation the method uses a convexity splitting scheme to discretize in the temporal variable and a nonconforming finite element method to discretize in the spatial variable and the scheme can preserve the mass conservation and energy dissipation properties of the original problem some typical phase transition phenomena are also observed through the numerical examples

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

implicit explicit integral deferred correction methods for stiff problems the main goal of this paper is to investigate the order reduction phenomenon that appears in the integral deferred correction indc methods based on implicit explicit imex runge kutta r k sch

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

superconvergence of the local discontinuous galerkin method for elliptic problems on cartesian grids in this paper we present a superconvergence result for the local discontinuous galerkin ldg method for a model elliptic problem on cartesian grids we identify a special numerical flux for which the l2 norm of the gradient and the l2 norm of the potential are of orders k 1 2 and k 1 respectively when tensor product polynomials of degree at most k are used for arbitrary meshes this special ldg method gives only the orders of convergence of k and k 1 2 respectively we present a series of numerical examples which establish the sharpness of our theoretical results

a new error analysis for discontinuous finite element methods for linear elliptic problems the standard a priori error analysis of discontinuous galerkin methods requires additional regularity on the solution of the elliptic boundary value problem in order to justify the galerkin orthogonality and to handle the normal derivative on element interfaces that appear in the discrete energy norm in this paper a new error analysis of discontinuous galerkin methods is developed using only the h k weak formulation of a boundary value problem of order 2k this is accomplished by replacing the galerkin orthogonality with estimates borrowed from a posteriori error analysis and by using a discrete energy norm that is well defined for functions in h k

on cotriangular hopf algebras in an earlier paper we proved that any triangular semisimple hopf algebra over an algebraically closed field k of characteristic 0 is obtained from the group algebra k g of a finite group g by twisting its comultiplication by a twist in the sense of drinfeld in this paper we generalize this result to not necessarily finite dimensional cotriangular hopf algebras namely our main result says that a cotriangular hopf algebra a over k is obtained from a function algebra of a proalgebraic group by twisting its multiplication by a hopf 2 cocycle and possibly changing its r form by a central grouplike element of a of order 2 if and only if the trace of the squared antipode on any finite dimensional subcoalgebra of a is the dimension of this subcoalgebra the generalization like the original theorem is proved using deligne s theorem on tannakian categories we then give examples of twisted function algebras and in particular show that in the infinite dimensional case the squared antipode may not equal the identity on the other hand we show that in all of our examples the squared antipode is unipotent and conjecture it to be the case for any twisted function algebra we prove this conjecture in a large number of special cases using the quantization theory of the first author and d kazhdan

convergence of a continuous galerkin method with mesh modification for nonlinear wave equations we consider space time continuous galerkin methods with mesh modification in time for semilinear second order hyperbolic equations we show a priori estimates in the energy norm without mesh conditions under reasonable assumptions on the choice of the spatial mesh in each time step we show optimal order convergence rates estimates of the jump in the riesz projection in two successive time steps are also derived

simple lie color algebras of witt type abstract let k be a field and let e k be a bicharacter defined on the multiplicative group we suppose that a is a graded associative k algebra that is color commutative with respect to e furthermore let be a nonzero graded k vector space of color derivations of a and suppose that is also color commutative with respect to the bicharacter e then with a rather natural definition a k a becomes a lie color algebra and we obtain necessary and sufficient conditions here for this lie color algebra to be simple with two minor exceptions when dim k 1 simplicity occurs if and only if a is graded simple and a a acts faithfully as color derivations on a

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

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

a hybrid high order method for the cahn hilliard problem in mixed form in this work we develop a fully implicit hybrid high order algorithm for the cahn hilliard problem in mixed form the space discretization hinges on local reconstruction operators from hybrid polynomial unknowns at elements and faces the proposed method has several advantageous features i it supports fairly general meshes possibly containing polyhedral elements and nonmatching interfaces ii it allows arbitrary approximation orders and iii it has a moderate computational cost thanks to the possibility of locally eliminating element based unknowns by static condensation we perform a detailed stability and convergence study proving optimal convergence rates in energy like norms numerical validation is also provided using some of the most common tests in the literature

landau levels on the hyperbolic plane in the presence of aharonov bohm fields abstract we consider the magnetic schrodinger operators on the poincare upper half plane with constant gaussian curvature 1 we assume the magnetic field is given by the sum of a constant field and the dirac measures placed on some lattice we give a sufficient condition for each landau level to be an infinitely degenerated eigenvalue we also prove the lowest landau level is not an eigenvalue if the above condition fails in particular the infinite degeneracy of the lowest landau level is equivalent to the infiniteness of the zero modes of the two dimensional pauli operator

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

connected braided hopf algebras abstract we prove a braided version of kostant cartier milnor moore theorem the category of connected cocommutative 2 id braided hopf algebras over a field of zero characteristic is equivalent via the enveloping construction to the category of generalized lie algebras this statement includes an embedding of any generalized lie algebra into its universal enveloping algebra the isomorphism h u prim h and primitive generation results the embedding theorem may be derived from the poincare birkhoff witt theorem for quadratic algebras of koszul type remark c we also provide a direct proof that uses neither koszul cohomologies nor the algebraic deformation theory we consider the primitive generation problem for subalgebras biideals and homomorphic images of connected braided hopf algebras in much more general context when the braiding is not necessary involutive and even it is not necessary invertible

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

fully discrete dynamic mesh discontinuous galerkin methods for the cahn hilliard equation of phase transition fully discrete discontinuous galerkin methods with variable meshes in time are developed for the fourth order cahn hilliard equation arising from phase transition in materials science the methods are formulated and analyzed in both two and three dimensions and are proved to give optimal order error bounds this coupled with the flexibility of the methods demonstrates that the proposed discontinuous galerkin methods indeed provide an efficient and viable alternative to the mixed finite element methods and non conforming plate finite element methods for solving fourth order partial differential equations

a nonconforming finite element method for the stationary navier stokes equations approximations to solutions of the inhomogeneous boundary value problem for the navier stokes equations are constructed via a nonstandard finite element method the velocity field is approximated using piecewise solenoidal functions that are totally discontinuous across interelement boundaries but which are pointwise divergence free on each element the pressure is approximated by c0 functions optimal rates of convergence results are obtained requiring only local quasi uniformity assumptions on the meshes

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

spectral properties of schrodinger operators with magnetic fields for a spin 12 particle abstract we discuss the spectral properties of schrodinger operators with magnetic fields especially for a spin 1 2 particle we are interested in the essential spectrum and the discrete spectrum we classified the spectrum by the asymptotic behavior of the magnetic field at infinity our main tools are the supersymmetry and the complex analysis in several variables we show that 0 is an eigenvalue with infinite multiplicity under suitable conditions

radford s s 4 formula for co frobenius hopf algebras abstract this note extends radford s formula for the fourth power of the antipode of a finite dimensional hopf algebra to co frobenius hopf algebras and studies equivalent conditions to a hopf algebra being involutory for finite dimensional and co frobenius hopf algebras

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

numerical analysis of a continuum model of phase transition a method for solution of the cahn hilliard equation is presented unlike previous work the discrete equations for the new method possess a lyapunov function this makes it possible to prove convergence of the approximate solutions without assumptions beyond those necessary for existence and uniqueness of the differential equation several consequences are explored

spectral properties of pauli operators on the poincare upper half plane we investigate the essential spectrum of the pauli operators and the dirac and the schrodinger operators with magnetic fields on the poincare upper half plane the magnetic fields under consideration are asymptotically constant which may be equal to zero or diverge at infinity moreover the aharonov casher type result is also considered

long time numerical simulations for phase field problems using p adaptive spectral deferred correction methods a high order and energy stable scheme is developed to simulate phase field models by combining the semi implicit spectral deferred correction sdc method and the energy stable convex splitting technique the convex splitting scheme we use here is a linear unconditionally stable method but is only of first order accuracy so the sdc method can be used to iteratively improve the rate of convergence however it is found that the accuracy improvement may affect the overall energy stability which is intrinsic to the phase field models to compromise the accuracy and stability a local p adaptive strategy is proposed to enhance the accuracy by sacrificing some local energy stability in an acceptable level the proposed strategy is found very useful for producing accurate numerical solutions at small time dynamics as well as long time steady state with reasonably large time stepsizes numerical experiments are carried out to demonstrate the high effectiveness of the proposed numerical strategy

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

finite element methods for the stochastic allen cahn equation with gradient type multiplicative noise this paper studies finite element approximations of the stochastic allen cahn equation with gradient type multiplicative noise that is white in time and correlated in space the sharp interface limit as the diffuse interface thickness vanishes of the stochastic allen cahn equation is formally a stochastic mean curvature flow which is described by a stochastically perturbed geometric law of the deterministic mean curvature flow two fully discrete finite element methods which are based on different time stepping strategies for the nonlinear term are proposed strong convergence with sharp rates for both fully discrete finite element methods is proved this is done with the crucial help of the holder continuity in time with respect to the spatial l 2 norm and h 1 seminorm for the strong solution of the stochastic allen cahn equation it also relies on the fact that high moments of the strong solution are bounded in various spatial and temporal norms numerical experiments are provided to gauge th

mixed finite element methods for the fully nonlinear monge ampere equation based on the vanishing moment method this paper studies mixed finite element approximations of the viscosity solution to the dirichlet problem for the fully nonlinear monge ampere equation det d 2u 0 f 0 based on the vanishing moment method which was proposed recently by the authors in x feng and m neilan j scient comp doi 10 1007 s10915 008 9221 9 2008 in this approach the second order fully nonlinear monge ampere equation is approximated by the fourth order quasilinear equation varepsilon delta 2 u varepsilon det d 2u varepsilon f it was proved in x feng trans ams submitted that the solution u varepsilon converges to the unique convex viscosity solution u 0 of the dirichlet problem for the monge ampere equation this result then opens a door for constructing convergent finite element methods for the fully nonlinear second order equations a task which has been impracticable before the goal of this paper is threefold first we develop a family of hermann miyoshi type mixed finite element methods for approximating the solution u varepsilon of the regularized fourth order problem which computes simultaneously u varepsilon and the moment tensor sigma varepsilon d 2u varepsilon second we derive error estimates which track explicitly the dependence of the error constants on the parameter varepsilon for the errors u varepsilon u varepsilon h and sigma 0 sigma h varepsilon finally we present a detailed numerical study on the rates of convergence in terms of powers of varepsilon for the error u 0 u h varepsilon and sigma varepsilon sigma h varepsilon and numerically examine what is the best mesh size h in relation to varepsilon in order to achieve these rates due to the strong nonlinearity of the underlying equation the standard perturbation argument for error analysis of finite element approximations of nonlinear problems does not work for the problem to overcome the difficulty we employ a fixed point technique which strongly relies on the stability of the linearized problem and its mixed finite element approximations

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

regular article calculation of two phase navier stokes flows using phase field modeling phase field models provide a way to model fluid interfaces as having finite thickness this can allow the computation of interface movement and deformation on fixed grids this paper applies phase field modeling to the computation of two phase incompressible navier stokes flows the navier stokes equations are modified by the addition of the continuum forcing email protected f where c is the composition variable and f is c s chemical potential the equation for interface advection is replaced by a continuum advective diffusion equation with diffusion driven by c s chemical potential gradients the paper discusses how solutions to these equations approach those of the original sharp interface navier stokes equations as the interface thickness e and the diffusivity both go to zero the basic flow physics of phase field interfaces is discussed straining flows can thin or thicken an interface and this must be resisted by a high enough diffusion on the other hand too large a diffusion will overly damp the flow these two constraints result in an upper bound for the diffusivity of o e and a lower bound of o e 2 within these two bounds the phase field navier stokes equations appear to generate an o e error relative to the exact sharp interface equations an o h 2 e 2 numerical method is introduced that is energy conserving in the sense that creation of interface energy by convection is always balanced by an equal decrease in kinetic energy caused by surface tension forcing an o h 4 e 4 compact scheme is introduced that takes advantage of the asymptotic comparatively smooth behavior of the chemical potential for o e accurate phase field models the optimum path to convergence for this scheme appears to be email protected 4 5 the asymptotic rate of convergence corresponding to this is o h 4 5 but results at practical resolutions show that the practical convergence of the method is generally considerably faster than linear extensive analysis and computations show that this scheme is very effective and accurate it allows the accurate calculation of two phase flows with interfaces only two cells wide computational results are given for linear capillary waves and for rayleigh taylor instabilities the first set of computations is compared to exact solutions of the diffuse interface equations and of the original sharp interface equations the rayleigh taylor computations test the ability of the method to compute highly deforming flows these flows include near singular phenomena such as interface coalescences and breakups contact line movement and the formation and breakup of thin wall films grid refinement studies are made and rapid convergence is found for macroscopic flow features such as instability growth rate and propagation speed wavelength and the general physical characteristics of the instability and mass transfer rates

pauli operator and aharonov casher theorem for measure valued magnetic fields we define the two dimensional pauli operator and identify its core for magnetic fields that are regular borel measures the magnetic field is generated by a scalar potential hence we bypass the usual a l 2 loc condition on the vector potential which does not allow to consider such singular fields we extend the aharonov casher theorem for magnetic fields that are measures with finite total variation and we present a counterexample in case of infinite total variation one of the key technical tools is a weighted l 2 estimate on a singular integral operator

semi implicit spectral deferred correction methods for highly nonlinear partial differential equations abstract the goal of this paper is to develop a novel semi implicit spectral deferred correction sdc time marching method the method can be used in a large class of problems especially for highly nonlinear ordinary differential equations odes without easily separating of stiff and non stiff components which is more general and efficient comparing with traditional semi implicit sdc methods the proposed semi implicit sdc method is based on low order time integration methods and corrected iteratively the order of accuracy is increased for each additional iteration and we also explore its local truncation error analytically this sdc method is intended to be combined with the method of lines which provides a flexible framework to develop high order semi implicit time marching methods for nonlinear partial differential equations pdes in this paper we mainly focus on the applications of the nonlinear pdes with higher order spatial derivatives e g convection diffusion equation the surface diffusion and willmore flow of graphs the cahn hilliard equation the cahn hilliard brinkman system and the phase field crystal equation coupled with the local discontinuous galerkin ldg spatial discretization the fully discrete schemes are all high order accurate in both space and time and stable numerically with the time step proportional to the spatial mesh size numerical experiments are carried out to illustrate the accuracy and capability of the proposed semi implicit sdc method

on the antipode of a quasitriangular hopf algebra let a r be a quasitriangular hopf algebra with antipode s in the category of vector spaces over a field k in this paper we show that s2 is inner which means s is bijective and when a is finite dimensional there is a grouplike element h e a such that s4 a h r for all a e a suppose that a is any hopf algebra with antipode s over the field k proposition 1 of section 1 gives a sufficient condition for 3 to be inner which we apply to quasitriangular hopf algebras in section 2 its statement and proof are based on calculations found in 2 pp 66671 generally s2 need not be inner suppose that a is finite dimensional if s2 were inner then all ideals of a would be invariant under s2 and hence all subcoalgebras of the dual hopf algebra a would be invariant under s where s s is the antipode of a there are examples 5 over algebraically closed lields in which the latter is not the case section 2 begins with a discussion of properties of quasit angular hopf algebras a r over the held k used in this paper write r cr c r ea a and set u cs r r the main result of section 2 is that u is invertible and s2 a uau for all a e a this result was established in z exercise 7 3 61 under the hypothesis that s is bijective if a is any finite dimensional hopf algebra with antipode s over the field k there are distinguished grouplike elements g e a and tl e a which relate left and right integrals and which together describe s4 in section 3 we consider s4 when a r is a unite dimensional quasit angular hopf algebra let h iiu where u c s r r l is defined as above and u s u i by relating h to g and 1 we are able to show that h is a grouplike element and that s4 a hah for all a e a when a is unimodular we prove that h g thus the product vu does not depend on r in the unimodular case we assume that the reader has some familiarity with the elementary aspects of the theory of hopf algebras a recommended reference is 6

discontinuous galerkin finite element method for the fourth order obstacle problem in this paper we consider the discontinuous galerkin finite element method for the fourth order obstacle problem which is the variational inequality problem of the first kind by defining the finite element approximation solution we obtain the optimal error estimate

the essential spectrum of schrodinger operators with asymptotically constant magnetic fields on the poincare upper half plane we study the essential spectrum of the magnetic schrodinger operators on the poincare upper half plane and establish a hyperbolic analog of iwatsuka s result j math kyoto univ 23 3 475 480 1983 on the stability of the essential spectrum under perturbations from constant magnetic fields

discontinuous galerkin methods for mass transfer through semipermeable membranes a discontinuous galerkin dg method for the numerical solution of initial boundary value multicompartment partial differential equation models interconnected with interface conditions is presented and analyzed the study of interface problems is motivated by models of mass transfer of solutes through semipermeable membranes more specifically a model problem consisting of a system of semilinear parabolic advection diffusion reaction partial differential equations in each compartment equipped with respective initial and boundary conditions is considered nonlinear interface conditions modeling selective permeability congestion and partial reflection are applied to the compartment interfaces an interior penalty dg method is presented for this problem and it is analyzed in the space discrete setting the a priori analysis shows that the method yields optimal a priori bounds provided the exact solution is sufficiently smooth numerical experiments indicate agreement with the theoretical bounds and highlight the stability of the numerical method in the advection dominated regime