a moving mesh method for the solution of the one dimensional phase field equations a moving mesh method is developed for the numerical solution of one dimensional phase change problems modelled by the phase field equations the computational mesh is obtained by equidistribution of a monitor function tailored for the functional variation of the phase field in the interfacial region existence and uniqueness of the discretised equations using a moving mesh are also established numerical results are given for classical and modified stefan test problems the numerical algorithm is relatively simple and is shown to be far more efficient than fixed grid methods

exponential codimension growth of pi algebras an exact estimate abstract letabe an associative pi algebra over a fieldfof characteristic zero by studying the exponential behavior of the sequence of codimensions cn a ofa we prove thatinv a limn c n a always exists and is an integer we also give an explicit way for computing such integer letbbe a finite dimensionalz2 graded algebra whose grassmann envelopeg b satisfies the same identities ofa theninv a inv g b dim c 0 dim c 1 wherec 0 c 1 is a suitablez2 graded semisimple subalgebra ofb

quantitative phase field modeling of dendritic growth in two and three dimensions we report the results of quantitative phase field simulations of the dendritic crystallization of a pure melt in two and three dimensions these simulations exploit a recently developed thin interface limit of the phase field model a karma and w j rappel phys rev e 53 r3017 1996 which is given here a detailed exposition this limit makes it possible to perform efficient computations with a smaller ratio of capillary length to interface thickness and with an arbitrary interface kinetic coefficient simulations in one and two dimensions are first carried out to test the accuracy of phase field computations performed within this limit dendrite tip velocities and tip shapes are found to be in excellent quantitative agreement with exact numerical benchmarks of solvability theory obtained by a boundary integral method both with and without interface kinetics simulations in three dimensions exploit in addition to the asymptotics a methodology to calculate grid corrections due to the surface tension and kinetic anisotropies they are used to test basic aspects of dendritic growth theory that pertain to the selection of the operating state of the tip and to the three dimensional morphology of needle crystals without sidebranches for small crystalline anisotropy simulated values of ensuremath sigma are slightly larger than solvability theory predictions computed by the boundary integral method assuming an axisymmetric shape and agree relatively well with experiments for succinonitrile given the uncertainty in the measured anisotropy in contrast for large anisotropy simulated ensuremath sigma values are significantly larger than the predicted values this disagreement however does not signal a breakdown of solvability theory it is consistent with the finding that the amplitude of the mathrm cos 4 ensuremath varphi mode which measures the departure of the tip morphology from a shape of revolution increases with anisotropy this departure can therefore influence the tip selection in a way that is not accurately captured by the axisymmetric approximation for large anisotropy finally the tail shape at a distance behind the tip that is large compared to the diffusion length is described by a linear law r ensuremath sim z with a slope dr dz that is nearly equal to the ratio of the two dimensional and three dimensional steady state tip velocities furthermore the evolution of the cross section of a three dimensional needle crystal with increasing distance behind the tip is nearly identical to the evolution of a two dimensional growth shape in time in accord with the current theory of the three dimensional needle crystal shape

a kam theorem for hamiltonian partial differential equations in higher dimensional spaces in this paper we give a kam theorem for a class of infinite dimensional nearly integrable hamiltonian systems the theorem can be applied to some hamiltonian partial differential equations in higher dimensional spaces with periodic boundary conditions to construct linearly stable quasi periodic solutions and its local birkhoff normal form the applications to the higher dimensional beam equations and the higher dimensional schrodinger equations with nonlocal smooth nonlinearity are also given in this paper

a two dimensional adaptive pseudo spectral method we develop a two dimensional adaptive pseudo spectral procedure which is capable of improving the approximation of functions which are rapidly varying in two dimensions the method is based on introducing two dimensional coordinate transformations chosen to minimize certain functionals of the solution to be approximated the method is illustrated by numerical computation of the solutions to a system of reaction diffusion equations modeling the gasless combustion of a solid fuel spatio temporal patterns are computed as a parameter related to the activation energy is increased above a critical value c the spatial patterns are characterized by a very rapid variation in the direction of the axis of the cylinder together with a standing wave pattern in the direction of the azimuthal angle for small values of c the solutions exhibit a nearly sinusoidal dependence in both time and as is increased further relaxation oscillations in both time and occur beyond a critical value of stable time periodic solutions are no longer found and the solution exhibits a quasi periodic time dependence

adaptive spectral methods with application to mixing layer computations this paper reports some experiments on the use of adaptive chebyshev pseudospectral methods for compressible mixing layer computations different functionals measuring the optimality of the polynomial approximation are discussed and compared in particular we address the problem of the practical computation of the various functionals the utility of the self adaptive method is then demonstrated by some examples from compressible mixing layer calculations

applications of hook young diagrams to p i algebras abstract a the multiplicities m in the cocharacter n a of any p i algebra a are polynomially bounded b a hook containing n a b is obtained from the hooks containing n a and n b these results are obtained by applying a theory of hook young diagrams to p i algebras and they generalize results known for algebras satisfying capelli identities

an analytic approximation to the cardinal functions of gaussian radial basis functions on an infinite lattice gaussian radial basis functions rbfs have been very useful in computer graphics and for numerical solutions of partial differential equations where these rbfs are defined on a grid with uniform spacing h as translates of the master function f x a h exp a 2 h 2 x 2 where a is a user choosable constant unfortunately computing the coefficients of f x jh a h requires solving a linear system with a dense matrix it would be much more efficient to rearrange the basis functions into the equivalent lagrangian or cardinal basis because the interpolation matrix in the new basis is the identity matrix the cardinal basis c j x a h is defined by the set of linear combinations of the gaussians such that c j kh 1 when k j and c j kh 0 for all integers k j we show that the cardinal functions for the uniform grid are c j x h a c x h j a where c x a a 2 p sin px sinh a 2x the relative error is only about 4exp email protected 2 a 2 as demonstrated by the explicit second order approximation it has long been known that the error in a series of gaussian rbfs does not converge to zero for fixed a as h 0 but only to an error saturation proportional to exp email protected 2 a 2 because the error in our approximation to the master cardinal function c x a is the square of the error saturation there is no penalty for using our new approximations to obtain matrix free interpolating rbf approximations to an arbitrary function f x the master cardinal function on a uniform grid in d dimensions is just the direct product of the one dimensional cardinal functions thus in two dimensions c x y a a 4 p 2 sin px sin py sinh a 2x sinh a 2y we show that the matrix free interpolation can be extended to non uniform grids by a smooth change of coordinates

kam tori of hamiltonian perturbations of 1d linear beam equations abstract in this paper one dimensional 1d nonlinear beam equations u tt u xxxx mu f u with hinged boundary conditions are considered the nonlinearity f is an analytic odd function and f u o u3 it is proved that for all real parameters m 0 but a set of small lebesgue measure the above equation admits small amplitude quasi periodic solutions corresponding to finite dimensional invariant tori of an associated infinite dimensional dynamical system the proof is based on infinite dimensional kam theory developed by kuksin lecture notes in mathematics vol 1556 springer berlin 1993 wayne commun math phys 127 1990 479 528 poschel ann sc norm sup pisa cl sci 23 1996 119 148

three dimensional adaptive grid method a three dimensional solution adaptive grid scheme is described which is suitable for complex fluid flows this method using tension and torsion spring analogies was previously developed and successfully applied for two dimensional flows in the present work a collection of three dimensional flow fields are used to demonstrate the feasibility and versatility of this concept to include an added dimension flow fields considered include 1 supersonic flow past an aerodynamic afterbody with a propulsive jet at incidence to the free stream 2 supersonic flow past a blunt fin mounted on a solid wall and 3 supersonic flow over a bump in addition to generating three dimensional solution adapted grids the method can also be used effectively as an initial grid generator the utility of the method lies in 1 optimum distribution of discrete grid points 2 improvement of accuracy 3 improved computational efficiency 4 minimization of data base sizes and 5 simplified three dimensional grid generation

moving mesh methods based on moving mesh partial differential equations several versions of a moving mesh method are developed based on a mesh spatial smoothing technique and on the moving mesh pdes derived in a previous paper these versions are quite simple and easy to program they are applied to three bench mark one dimensional problems which show different solution behaviour the numerical results clearly demonstrate that the present methods are capable of accurately tracking rapid spatial and temporal transitions

an adaptive meshfree method for phase field models of biomembranes part i we present an adaptive meshfree method for phase field models of biomembranes max ent smooth approximants can deal with the high order phase field equations the method resolves adaptively the models at an affordable computational cost convergence to sharp interface limit is shown with small regularization parameters we present an adaptive meshfree method to approximate phase field models of biomembranes in such models the helfrich curvature elastic energy the surface area and the enclosed volume of a vesicle are written as functionals of a continuous phase field which describes the interface in a smeared manner such functionals involve up to second order spatial derivatives of the phase field leading to fourth order euler lagrange partial differential equations pde the solutions develop sharp internal layers in the vicinity of the putative interface and are nearly constant elsewhere thanks to the smoothness of the local maximum entropy max ent meshfree basis functions we approximate numerically this high order phase field model with a direct ritz galerkin method the flexibility of the meshfree method allows us to easily adapt the grid to resolve the sharp features of the solutions thus the proposed approach is more efficient than common tensor product methods e g finite differences or spectral methods and simpler than unstructured c 0 finite element methods applicable by reformulating the model as a system of second order pde the proposed method implemented here under the assumption of axisymmetry allows us to show numerical evidence of convergence of the phase field solutions to the sharp interface limit as the regularization parameter approaches zero in a companion paper we present a lagrangian method based on the approximants analyzed here to study the dynamics of vesicles embedded in a viscous fluid

adaptive mesh strategies for the spectral element method abstract an adaptive spectral element method has been developed for the efficient solution of time dependent partial differential equations adaptive mesh strategies that include resolution refinement and coarsening by three different methods are illustrated on solutions to the one dimensional viscous burgers equation and the two dimensional navier stokes equations for driven flow in a cavity sharp gradients singularities and regions of poor resolution are resolved optimally as they develop in time using error estimators which indicate the choice of refinement to be used the adaptive formulation presents significant increases in efficiency flexibility and general capabilities for high order spectral methods

on balanced moving mesh methods moving mesh methods are a widely used approach in the numerical solution of pdes where the original pdes are transformed from a physical domain to a computational domain the objective is to utilize a uniform mesh in the computational domain to get a non uniform physical mesh that better captures the behavior of the solution the movement of the physical mesh points can be governed by a moving mesh pde associated with a corresponding monitor function and both the original pdes and the moving mesh pdes must be solved simultaneously the motivation for this paper is to study a balanced moving mesh method where the aim is to strike a balance between the properties of the solution of the original pde and that of the moving mesh pde we focus on particular choices of the monitor function that give both a well behaved mesh transformation and a well behaved solution in the computational domain both theoretical analysis and numerical experiments are presented as illustrations

long time stability of small finite gap solutions of the cubic nonlinear schrodinger equation on t2 abstract in this paper we study long time stability of a class of nontrivial quasi periodic solutions depending on one spacial variable of the cubic defocusing non linear schrodinger equation on the two dimensional torus we prove that these quasi periodic solutions are orbitally stable for finite but long times provided that their fourier support and their frequency vector satisfy some complicated but explicit condition which we show holds true for most solutions the proof is based on a normal form result more precisely we expand the hamiltonian in a neighborhood of a quasi periodic solution we reduce its quadratic part to diagonal constant coefficients through a kam scheme and finally we remove its cubic terms with a step of nonlinear birkhoff normal form the main difficulty is to impose second and third order melnikov conditions this is done by combining the techniques of reduction in order of pseudo differential operators with the algebraic analysis of resonant quadratic hamiltonians

practical aspects of formulation and solution of moving mesh partial differential equations moving mesh partial differential equations mmpdes are used in the mmpde moving mesh method to generate adaptive moving meshes for the numerical solution of time dependent problems how mmpdes are formulated and solved is crucial to the efficiency and robustness of the method in this paper several practical aspects of formulating and solving mmpdes are studied they include spatial balance scaling invariance effective control of mesh concentration bounds on time steps multiple sub steps and two level mesh movement numerical results are also given c 2001 academic press

numerical solution of the quasilinear poisson equation in a nonuniform triangle mesh a finite difference method using a nonuniform triangle mesh is described for the numerical solution of the nonlinear two dimensional poisson equation s 0 where is a function of or its derivatives sis a function of position and or its normal derivative is specified on the boundary the finite difference equations are solved by successive overrelaxation the triangle mesh which is constructed numerically by solving laplace s equation is easily adapted to nonrectangular boundaries and interfaces examples of numerical results are given for the magnetostatic problem with iron and other possible applications are mentioned

simple adaptive grids for 1 d initial value problems abstract in solving 1 d initial value problems by implicit finite difference methods it is advantageous to use an adaptive grid to provide local regions of high resolution and allow larger time steps to be taken a simple method of constructing such grids is presented and the benefits are demonstrated by calculations of sod s shock tube problem g a sod j comput phys 27 1 1978 and of a supernova explosion

moving mesh strategy based on a gradient flow equation for two dimensional problems in this paper we introduce a moving mesh method for solving pdes in two dimensions it can be viewed as a higher dimensional generalization of the moving mesh pde mmpde strategy developed in our previous work for one dimensional problems w huang y ren and r d russell siam j numer anal 31 1994 pp 709 730 the mmpde is derived from a gradient flow equation which arises using a mesh adaptation functional in turn motivated from the theory of harmonic maps geometrical interpretations are given for the gradient equation and functional and basic properties of this mmpde are discussed numerical examples are presented where the method is used both for mesh generation and for solving time dependent pdes the results demonstrate the potential of the mesh movement strategy to concentrate the mesh points so as to adapt to special problem features and to also preserve a suitable level of mesh orthogonality

pseudospectral solution of near singular problems using numerical coordinate transformations based on adaptivity the work presented here describes a method of coordinate transformation that enables spectral methods to be applied efficiently to differential problems with steep solutions the approach makes use of the adaptive finite difference method presented by huang and sloan siam j sci comput 15 1994 pp 776 797 this method is applied on a coarse grid to obtain a rough approximation of the solution and a suitable adapted mesh the adaptive finite difference solution permits the construction of a smooth coordinate transformation that relates the computational space to the physical space the map between the spaces is based on chebyshev polynomial interpolation finally the standard pseudospectral ps method is applied to the transformed differential problem to obtain highly accurate nonoscillatory numerical solutions numerical results are presented for steady problems in one and two space dimensions

sparse low energy decomposition for viscous conservation laws for viscous conservation laws solutions contain smooth but high contrast features which require the use of fine grids to properly resolve on coarse grids these high contrast jumps resemble shocks rather than their true viscous profiles which could lead to issues in the numerical approximation of their underlying dynamics in many cases the equations of motion emit traveling wave solutions which can be used to represent the viscous profiles analytically the traveling wave solutions can be thought of as a lower dimensional representation of the motion since they contain information from the evolution equation but are constant along certain time space curves using a parameterized basis involving the traveling waves along with the sparse low energy decompositions found in imaging sciences we propose an approximation to viscous conservation laws which separates the coarse smooth component from the sharp fine one our method provides an appropriate approximation to the solution on a coarse grid thereby accurately under resolving the viscous profile this is similar to the philosophy of shock capturing methods in the sense that we want to capture the viscous front without needing to resolve the profile theoretical results on the consistency of our method are shown in general we provide several computational examples for convex and non convex fluxes

a conservative numerical method for the cahn hilliard equation with dirichlet boundary conditions in complex domains in this paper we present a conservative numerical method for the cahn hilliard equation with dirichlet boundary conditions in complex domains the method uses an unconditionally gradient stable nonlinear splitting numerical scheme to remove the high order time step stability constraints the continuous problem has the conservation of mass and we prove the conservative property of the proposed discrete scheme in complex domains we describe the implementation of the proposed numerical scheme in detail the resulting system of discrete equations is solved by a nonlinear multigrid method we demonstrate the accuracy and robustness of the proposed dirichlet boundary formulation using various numerical experiments we numerically show the total energy decrease and the unconditionally gradient stability in particular the numerical results indicate the potential usefulness of the proposed method for accurately calculating biological membrane dynamics in confined domains

some results concerning the multiplicities of cocharacters of superalgebras with graded involution abstract let a be a finitely generated superalgebra with graded involution over a field f of characteristic zero and let n 1 n 4 a n 1 n 4 n be the n 1 n 4 cocharacter of a in this paper we present some results concerning the multiplicities of cocharacters for this kind of algebras

a characterization of p i algebras with bounded multiplicities of the cocharacters abstract we characterize varieties of p i algebras with bounded multiplicities of the cocharacters a variety is such if and only if it does not contain the upper triangular 2 2 matrices this also yields a characterization of the varieties with bounded colength

graded algebras with polynomial growth of their codimensions abstract let a be an algebra over a field of characteristic 0 and assume a is graded by a finite group g we study combinatorial and asymptotic properties of the g graded polynomial identities of a provided a is of polynomial growth of the sequence of its graded codimensions roughly speaking this means that the ideal of graded identities is very large we relate the polynomial growth of the codimensions to the module structure of the multilinear elements in the relatively free g graded algebra in the variety generated by a we describe the irreducible modules that can appear in the decomposition we show that their multiplicities are eventually constant depending on the shape obtained by the corresponding multipartition after removing its first row we relate moreover the polynomial growth to the colengths finally we describe in detail the algebras whose graded codimensions are of linear growth

kam for the derivative nonlinear schrodinger equation with periodic boundary conditions abstract this paper is concerned with the derivative nonlinear schrodinger equation with periodic boundary conditions i u t u x x i f u 2 u x 0 x t r 2 z where f is real analytic in some neighborhood of the origin in c f 0 0 and f 0 0 we show the above equation possesses cantor families of smooth quasi periodic solutions of small amplitude the proof is based on an infinite dimensional kam theorem for unbounded perturbation vector fields

efficient r adaptive isogeometric analysis with winslow s mapping and monitor function approach abstract motivated by the moving mesh method in conventional finite element analysis in this paper we propose an r adaptive isogeometric analysis framework by winslow s mapping and monitor function approach we construct a new monitor function based on the absolute curvature metric of the isogeometric solution surface in the proposed r adaptive framework the redistribution of the inner control points is performed to increase the density of control points in the regions where curvature based or error based monitor functions are maximum the new locations of the inner control points are obtained by solving governing pdes using isogeometric collocation method which is more efficient in solving large scale problems by employing winslow s mapping we ensure that the generated r adaptive parameterization is valid for isogeometric analysis and can be used as optimal initial grid for subsequent h refinements multiple two dimensional heat conduction examples are solved which confirm the effectiveness of the proposed method

moving mesh partial differential equations mmpdes based on the equidistribution principle this paper considers several moving mesh partial differential equations that are related to the equidistribution principle several of these are new and some correspond to discrete moving mesh equations that have been used by others their stability is analyzed and it is seen that a key term for most of these moving mesh pdes is a source like term that measures the level of equidistribution it is shown that under weak assumptions mesh crossing cannot occur for most of them finally numerical experiments for these various moving mesh pdes are performed to study their relative properties

an unconditionally stable hybrid numerical method for solving the allen cahn equation we present an unconditionally stable second order hybrid numerical method for solving the allen cahn equation representing a model for antiphase domain coarsening in a binary mixture the proposed method is based on operator splitting techniques the allen cahn equation was divided into a linear and a nonlinear equation first the linear equation was discretized using a crank nicolson scheme and the resulting discrete system of equations was solved by a fast solver such as a multigrid method the nonlinear equation was then solved analytically due to the availability of a closed form solution various numerical experiments are presented to confirm the accuracy efficiency and stability of the proposed method in particular we show that the scheme is unconditionally stable and second order accurate in both time and space

an efficient moving mesh spectral method for the phase field model of two phase flows we develop in this paper a moving mesh spectral method for the phase field model of two phase flows with non periodic boundary conditions the method is based on a variational moving mesh pde for the phase function coupled with efficient semi implicit treatments for advancing the mesh function the phase function and the velocity and pressure in a decoupled manner ample numerical results are presented to demonstrate the accuracy and effectiveness of the moving mesh spectral method

quasi periodic solutions for a completely resonant beam equation with a nonlinear term depending on the time and space variables abstract this article is devoted to the study of a completely resonant beam equation with an x periodic and t quasi periodic nonlinear term it is proved that the equation admits small amplitude linearly stable and quasi periodic solutions for most values of the frequency vector by utilizing the measure estimation of infinitely many small divisors we construct a real analytic symplectic change of coordinates which can transform the hamiltonian into some birkhoff normal form we show an infinite dimensional kam theorem for non autonomous beam equations which is applied to prove the existence of quasi periodic solutions

a robust moving mesh method for spectral collocation solutions of time dependent partial differential equations this work extends the machinery of the moving mesh partial differential equation mmpde method to the spectral collocation discretization of time dependent partial differential equations unlike previous approaches which bootstrap the moving grid from a lower order finite difference discretization this work uses a consistent spectral collocation discretization for both the grid movement problem and the underlying physical partial differential equation additionally this work develops an error monitor function based on filtering in the spectral domain which concentrates grid points in areas of locally poor resolution without relying on an assumption of locally steep gradients this makes the mmpde method more robust in the presence of rarefaction waves which feature rapid change in higher order derivatives

a semi analytical fourier spectral method for the allen cahn equation in recent years fourier spectral methods have been widely used as a powerful tool for solving phase field equations to improve its effectiveness many researchers have employed stabilized semi implicit fourier spectral sifs methods which allow a much larger time step than a usual explicit scheme our mathematical analysis and numerical experiments however suggest that an effective time step is smaller than a time step specified in the sifs schemes in consequence the sifs scheme is inaccurate for a considerably large time step and may lead to incorrect morphologies in phase separation processes in order to remove the time step constraint and guarantee the accuracy in time for a sufficiently large time step we present a first and a second order semi analytical fourier spectral safs methods for solving the allen cahn equation the core idea of the methods is to decompose the original equation into linear and nonlinear subequations which have closed form solutions in the fourier and physical spaces respectively both the first and the second order methods are unconditionally stable and numerical experiments demonstrate that our proposed methods are more accurate than the stabilized semi implicit fourier spectral method

an efficient dynamically adaptive mesh for potentially singular solutions abstract we develop an efficient dynamically adaptive mesh generator for time dependent problems in two or more dimensions the mesh generator is motivated by the variational approach and is based on solving a new set of nonlinear elliptic pdes for the mesh map when coupled to a physical problem the mesh map evolves with the underlying solution and maintains high adaptivity as the solution develops complicated structures and even singular behavior the overall mesh strategy is simple to implement avoids interpolation and can be easily incorporated into a broad range of applications the efficacy of the mesh is first demonstrated by two examples of blowing up solutions to the 2 d semilinear heat equation these examples show that the mesh can follow with high adaptivity a finite time singularity process the focus of applications presented here is however the baroclinic generation of vorticity in a strongly layered 2 d boussinesq fluid a challenging problem the moving mesh follows effectively the flow resolving both its global features and the almost singular shear layers developed dynamically the numerical results show the fast collapse to small scales and an exponential vorticity growth

spectral implementation of an adaptive moving mesh method for phase field equations phase field simulations have been extensively applied to modeling microstructure evolution during various materials processes however large scale simulations of three dimensional 3d microstructures are still computationally expensive among recent efforts to develop advanced numerical algorithms the semi implicit fourier spectral method is found to be particularly efficient for systems involving long range interactions as it is able to utilize the fast fourier transforms fft on uniform grids in this paper we report our recent progress in making grid points spatially adaptive in the physical domain via a moving mesh strategy while maintaining a uniform grid in the computational domain for the spectral implementation this approach not only provides more accurate treatment at the interfaces requiring higher resolution but also retains the numerical efficiency of the semi implicit fourier spectral method numerical examples using the new adaptive moving mesh semi implicit fourier spectral method are presented for both two and three space dimensional microstructure simulations and they are compared with those obtained by other methods by maintaining a similar accuracy the proposed method is shown to be far more efficient than the existing methods for microstructures with small ratios of interfacial widths to the domain size

on codimension growth of finitely generated associative algebras abstract letabe a pi algebra over a fieldf we study the asymptotic behavior of the sequence of codimensionscn a ofa we show that ifais finitely generated overftheninv a limn c n a always exists and is an integer we also obtain the following characterization of simple algebras ais finite dimensional central simple overfif and only ifinv a dim a

noncommutative invariant theory of symplectic and orthogonal groups abstract we present a method for computing the hilbert series of the algebra of invariants of the complex symplectic and orthogonal groups acting on graded noncommutative algebras with homogeneous components which are polynomial modules of the general linear group we apply our method to compute the hilbert series for different actions of the symplectic and orthogonal groups on the relatively free algebras of the varieties of associative algebras generated respectively by the grassmann algebra and the algebra of 2 2 upper triangular matrices these two varieties are remarkable with the property that they are the only minimal varieties of exponent 2

a three dimensional adaptive method based on the iterative grid redistribution in this paper we develop a three dimensional adaptive method based on iterative grid redistribution technique introduced in j comput phys 159 2000 246 the key step for the successful implementation is a fast algorithm for grid generation which is composed of solving the linear grid equation systems and an efficient method for inverting a map by computing iso surface intersections to carry out the three dimensional calculations the whole procedure is also parallelized on a pc cluster the improved and extended three dimensional adaptive method is applied to solve pdes with singular solutions that have three dimensional structures numerical experiments have demonstrated the method s effectiveness

a study of monitor functions for two dimensional adaptive mesh generation in this paper we study the problem of two dimensional adaptive mesh generation using a variational approach and specifically the effect that the monitor function has on the resulting mesh behavior the basic theoretical tools employed are green s function for elliptic problems and the eigendecomposition of symmetric positive definite matrices based upon this study a general strategy is suggested for how to choose the monitor function and numerical results are presented for illustrative purposes the three dimensional case is also briefly discussed it is noted that the strategy used here can be applied to other elliptic mesh generation techniques as well

an adaptive pseudo spectral method for reaction diffusion problems the spectral interpolation error was considered for both the chebyshev pseudo spectral and galerkin approximations a family of functionals i sub r u with the property that the maximum norm of the error is bounded by i sub r u j sub r where r is an integer and j is the degree of the polynomial approximation was developed these functionals are used in the adaptive procedure whereby the problem is dynamically transformed to minimize i sub r u the number of collocation points is then chosen to maintain a prescribed error bound the method is illustrated by various examples from combustion problems in one and two dimensions

a normal form for beam and non local nonlinear schrodinger equations we discuss normal forms of the completely resonant nonlinear beam equation and nonlinear schrodinger equation we work in n 1 spatial dimensions and study both periodic and dirichlet boundary conditions on cubes we discuss the applications to the problem of finding quasi periodic solutions in the case of periodic boundary and the dimension n 2 we apply kam theory and prove the existence and stability of quasi periodic solutions

nonlocal damage model using the phase field method theory and applications abstract a new nonlocal gradient based damage model is proposed for isotropic elastic damage using the phase field method in order to show the evolution of damage in brittle materials the general framework of the phase field model pfm is discussed and the order parameter is related to the damage variable in continuum damage mechanics cdm the time dependent ginzburg landau equation which is also termed the allen cahn equation is used to describe the damage evolution process specific length scale which addresses the interface region in which the process of changing undamaged solid to fully damaged material microcracks occurs is defined in order to capture the effect of the damaged localization zone a new implicit damage variable is proposed through the phase field theory details of the different aspects and regularization capabilities are illustrated by means of numerical examples and the validity and usefulness of the phase field modeling approach is demonstrated

efficient computation of dendritic growth with r adaptive finite element methods this paper deals with the application of a moving grid method to the solution of a phase field model for dendritic growth in two and three dimensions a mesh is found as the solution of an optimization problem that automatically includes the boundary conditions and is solved using a multi grid approach the governing equations are discretized in space by linear finite elements and a split time level scheme is used to numerically integrate in time one novel aspect of the method is the choice of a regularized monitor function the moving grid method enables us to obtain accurate numerical solutions with much less degree of freedoms it is demonstrated numerically that the tip velocity obtained by our method is in good agreement with the previously published results