conservation laws for conformally invariant variational problems we succeed in writing 2 dimensional conformally invariant non linear elliptic pde harmonic map equation prescribed mean curvature equations etc in divergence form these divergence free quantities generalize to target manifolds without symmetries the well known conservation laws for weakly harmonic maps into homogeneous spaces from this form we can recover without the use of moving frame all the classical regularity results known for 2 dimensional conformally invariant non linear elliptic pde see hel it enables us also to establish new results in particular we solve a conjecture by e heinz asserting that the solutions to the prescribed bounded mean curvature equation in arbitrary manifolds are continuous and we solve a conjecture by s hildebrandt hil1 claiming that critical points of continuously differentiable elliptic conformally invariant lagrangian in two dimensions are continuous

long time dynamics for semi relativistic nls and half wave in arbitrary dimension we consider the cauchy problems associated with semi relativistic nls snls and half wave hw in particular we focus on the following two main questions local global cauchy theory existence and stability instability of ground states in between other results we prove the existence and stability of ground states for snls in the l 2 supercritical regime this is in sharp contrast with the instability of ground states for the corresponding hw which is also established along the paper by showing an inflation of norms phenomenon concerning the cauchy theory we show under radial symmetry assumption the following results a local existence result in h 1 for energy subcritical nonlinearity and a global existence result in the l 2 subcritical regime

an adaptive splitting approach for the quenching solution of reaction diffusion equations over nonuniform grids the numerical solution of a nonlinear degenerate reaction diffusion equation of the quenching type is investigated while spatial derivatives are discretized over symmetric nonuniform meshes a peaceman rachford splitting method is employed to advance solutions of the semidiscretized system the temporal step is determined adaptively through a suitable arc length monitor function a criterion is derived to ensure that the numerical solution acquired preserves correctly the positivity and monotonicity of the analytical solution weak stability is proven in a von neumann sense via the norm computational examples are presented to illustrate our results

a moving finite element method with error estimation and refinement for one dimensional time dependent partial differential equations we discuss a moving finite element method for solving vector systems of time dependent partial differential equations in one space dimension the mesh is moved so as to equidistribute the spatial component of the discretization error in h 1 we present a method of estimating this error by using p hierarchic finite elements the error estimate is also used in an adaptive mesh refinement procedure to give an algorithm that combines mesh movement and refinement we discretize the partial differential equations in space using a galerkin procedure with piecewise linear elements to approximate the solution and quadratic elements to estimate the error a system of ordinary differential equations for mesh velocities are used to control element motions we use existing software for stiff ordinary differential equations for the temporal integration of the solution the error estimate and the mesh motion computational results using a code based on our method are presented for several examples

finite element method for one dimensional and two dimensional time dependent problems with b splines abstract one dimensional and two dimensional time dependent problems have been solved by the galerkin method with cubic b splines as basis functions the redefining of basis functions has been done for two dimensional problems for the dirichlet type boundary conditions resulting in a non homogeneous part in the approximation the equidistribution of the error principle given by carl de boor for one dimensional problems has been extended to two dimensional problems the solutions for nonlinear problems are obtained as the limit of solutions of a sequence of linear problems generated by the quasilinearization technique the method developed with these features compares favourably with the methods available in literature

sharp eigenvalue bounds and minimal surfaces in the ball we prove existence and regularity of metrics on a surface with boundary which maximize sigma 1 l where sigma 1 is the first nonzero steklov eigenvalue and l the boundary length we show that such metrics arise as the induced metrics on free boundary minimal surfaces in the unit ball b n for some n in the case of the annulus we prove that the unique solution to this problem is the induced metric on the critical catenoid the unique free boundary surface of revolution in b 3 we also show that the unique solution on the mobius band is achieved by an explicit s 1 invariant embedding in b 4 as a free boundary surface the critical mobius band for oriented surfaces of genus 0 with arbitrarily many boundary components we prove the existence of maximizers which are given by minimal embeddings in b 3 we characterize the limit as the number of boundary components tends to infinity to give the asymptotically sharp upper bound of 4 pi we also prove multiplicity bounds on sigma 1 in terms of the topology and we give a lower bound on the morse index for the area functional for free boundary surfaces in the ball

existence of infinitely many stationary solutions of the l 2 subcritical and critical nlse on compact metric graphs abstract we investigate the existence of stationary solutions for the nonlinear schrodinger equation on compact metric graphs in the l 2 subcritical setting we prove the existence of an infinite number of such solutions for every value of the mass in the critical regime the existence of infinitely many solutions is established if the mass is lower than a threshold value while global minimizers of the nls energy exist if and only if the mass is lower or equal to the threshold moreover the relation between this threshold and the topology of the graph is characterized the investigation is based on variational techniques and some new versions of gagliardo nirenberg inequalities

an effective z stretching method for paraxial light beam propagation simulations a z stretching finite difference method is developed for simulating the paraxial light beam propagation through a lens in a cylindrically symmetric domain by introducing a domain transformation in the z direction we solve the corresponding complex difference equations containing an interface singularity over a computational space for great simplicity and efficiency a specially designed matrix analysis is constructed to the study the numerical stability computational experiments are carried out for demonstrating our results

a particle method for collisional kinetic equations i basic theory and one dimensional results abstract a new method is introduced for describing the collisional term in the framework of a particle scheme the equations of motion of the particles contain an additional term which takes into account the collisions numerical techniques are shown to compute this extra term and to solve the equations exact results concerning stability and consistency of the schemes for the diffusion equation are derived some numerical results for the one dimensional case are reported

remarks on quenching on considere le probleme aux limites parabolique non lineaire suivant u t u f u t 0 x u 0 t 0 n u 0 x u 0 x 0 x on demontre un phenomene de trempe pour des non linearites non convexes dans un espace de dimension arbitraire

stability of moving mesh systems of partial differential equations moving mesh methods based on the equidistribution principle ep are studied from the viewpoint of stability of the moving mesh system of differential equations for fine spatial grids the moving mesh system inherits the stability of the original discretized partial differential equation pde unfortunately for some pdes the moving mesh methods require so many spatial grid points that they no longer appear to be practical failures and successes of the moving mesh method applied to three reaction diffusion problems are explained via an analysis of the stability and accuracy of the moving mesh pde

highly efficient oscillation free solution of the transport equation over long times and large spaces transport problems with sharp transitions or high convection often stress even the most advanced computational methods beyond feasibility limits one promising line of attack involves transforming problems to a moving deforming coordinate system mdcs using mdcs as recently developed one may concentrate computational attention when and where it is most appropriate also possibly greatly reducing the effective magnitude of terms which cause trouble in principle mdcs may be used with finite difference finite element or other numerical methods according to preference in the particular finite element system advanced here mesh movement may be variable in time and space so that physically fixed boundaries may remain fixed in the transformed coordinate system a simple finite difference system is used in time when the method is applied to the standard transport equation test case great efficiency and accuracy are obtained in early time even very steep fronts may be represented well with no oscillations using space and time step sizes well beyond conventional peclet courant number constraints more important problems are solved involving transport over much longer times and larger spaces when diffusion has some chance to operate and hence a parabolic model is more warranted the same accuracy and stability are achieved as during early time using no more nodes and time steps overall than during early time runs this constitutes orders of magnitude savings in effort over what the application of conventional constraints would require relative efficiency from application of mdcs to other problems would depend on the particulars of each case

a comparison of differential quadrature methods for the solution of partial differential equations abstract discrete ordinate methods for the solution of partial differential equations are examined and compared a novel approach for the discrete representation of differential operators based on split range polynomial expansions is introduced the utility of the method is demonstrated for the case of differentiation of functions involving steep gradients the solution of burgers equation is presented to illustrate the effectiveness of the technique for the solution of nonlinear partial differential equations exhibiting nearly discontinuous solutions comparisons are made with other differential quadrature methods

negative energy ground states for the l2 critical nlse on metric graphs we investigate the existence of ground states with prescribed mass for the focusing nonlinear schrodinger equation with l2 critical power nonlinearity on noncompact quantum graphs we prove that unlike the case of the real line for certain classes of graphs there exist ground states with negative energy for a whole interval of masses a key role is played by a thorough analysis of gagliardo nirenberg inequalities and by estimates of the optimal constants most of the techniques are new and suited to the investigation of variational problems on metric graphs

temporal derivatives in the finite element method on continuously deforming grids when solving certain time dependent partial differential equations using a finite element technique on a deforming grid it is shown that there is a need to differentiate the trial solution with respect to the position of each of the moveable node points a result is presented which enables these nonstandard derivatives to be expressed in terms of the standard spatial derivatives of the trial functions provided that certain conditions are satisfied these conditions are investigated and shown to be satisfied for a large class of trial spaces even when the necessary conditions are not satisfied by all of the trial functions being used the result is shown to be still of great use in evaluating these awkward nodal derivatives 1 u x t y u x t a u

energy identity for approximations of harmonic maps from surfaces we prove the energy identity for min max sequences of the sacks uhlenbeck and the biharmonic approximation of harmonic maps from surfaces into general target manifolds the proof relies on hopf differential type es timates for the two approximations and on estimates for the concentration radius of bubbles

two dimensional fully adaptive solutions of solid solid alloying reactions abstract solid solid alloying reactions occur in a variety of pyrotechnical applications they arise when a mixture of powders composed of appropriate oxidizing and reducing agents is heated the large quantity of heat evolved produces a self propagating reaction front that is often very narrow with sharp changes in both the temperature and the concentrations of the reacting species solution of problems of this type with an equispaced or mildly nonuniform grid can be extremely inefficient in this paper we develop a two dimensional fully adaptive method for solving problems of this class the method adaptively adjusts the number of grid points needed to equidistribute a positive weight function over a given mesh interval in each direction at each time level we monitor the solution from one time level to another to ensure that the local error per unit step associated with the time differencing method is below some specified tolerance the method is applied to several examples involving exothermic diffusion controlled self propagating reactions in packed bed reactors

a numerical study of three moving grid methods for one dimensional partial differential equations which are based on the method of lines in recent years several sophisticated packages based on the method of lines mol have been developed for the automatic numerical integration of time dependent problems in partial differential equations pdes notably for problems in one space dimension these packages greatly benefit from the very successful developments of automatic stiff ordinary differential equation solvers however from the pde point of view they integrate only in a semiautomatic way in the sense that they automatically adjust the time step sizes but use just a fixed space grid chosen a priori for the entire calculation for solutions possessing sharp spatial transitions that move e g travelling wave fronts or emerging boundary and interior layers a grid held fixed for the entire calculation is computationally inefficient since for a good solution this grid often must contain a very large number of nodes in such cases methods which attempt automatically to adjust the sizes of both the space and the time steps are likely to be more successful in efficiently resolving critical regions of high spatial and temporal activity methods and codes that operate this way belong to the realm of adaptive or moving grid methods following the mol approach this paper is devoted to an evaluation and comparison mainly based on extensive numerical tests of three moving grid methods for 1d problems viz the finite element method of miller and co workers the method published by petzold and a method based on ideas adopted from dorii and drury our examination of these three methods is aimed at assessing which is the most suitable from the point of view of retaining the acknowledged features of reliability robustness and efficiency of the conventional mol approach therefore considerable attention is paid to the temporal performance of the methods

an adaptive grid method for degenerate semilinear quenching problems abstract this paper studies the structure and implementation of a specially formulated adaptive grid method for computing the numerical solution of a class of degenerate semilinear quenching problems modified arc length procedures are used in both the space and time for robust adjustments of the spatial and temporal discretizations throughout the computation a milne like device is introduced in the error estimation and control of the computational error the numerical solution generated converges monotonically to the physical solution of the underlying problem up to the quenching time numerical examples are given for demonstrating our conclusions

a semi adaptive compact splitting method for the numerical solution of 2 dimensional quenching problems abstract this article studies a semi adaptive compact peaceman rachford splitting method for solving two dimensional nonlinear reaction diffusion equations with singular source terms while an adaption is utilized in the temporal direction uniform grids are considered in the space it is shown that the compact scheme is stable and convergent when its dimensional courant numbers are within the frame of a window determined by the given spatial domain though such a window implies a considerable restriction on the decomposed compact scheme the new computational strategy acquired is highly efficient and reliable for a variety of applications numerical examples are given to illustrate our conclusions and indicate that the method constructed is effective in determining key singularity characteristics such as the quenching time and critical domain

equidistributing principles in moving finite element methods abstract recently miller and his co workers proposed a moving finite element method based on a least squares principle this was followed by a similar method by the present authors using a petrov galerkin approach in this paper the two methods are compared in particular it is shown that both methods move their nodes according to an approximate equidistributing principle this observation leads to a criterion for the placement of the nodes it is also shown that the penalty function designed by miller may also be used with the petrov galerkin method finally numerical examples are given illustrating the performance of the two methods

bound states of the nls equation on metric graphs with localized nonlinearities abstract we investigate the existence of multiple bound states of prescribed mass for the nonlinear schrodinger equation on a noncompact metric graph the main feature is that the nonlinearity is localized only in a compact part of the graph our main result states that for every integer k the equation possesses at least k solutions of prescribed mass provided that the mass is large enough these solutions arise as constrained critical points of the nls energy functional estimates for the energy of the solutions are also established

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

solving degenerate reaction diffusion equations via variable step peaceman rachford splitting this paper studies the numerical solution of two dimensional nonlinear degenerate reaction diffusion differential equations with singular forcing terms over rectangular domains the equations considered may generate strong quenching singularities this investigation focuses on a variable time step peaceman rachford splitting method for the aforementioned problem the time adaptation is implemented based on arc length estimations of the first time derivative of the solution the two dimensional problem is split into several one dimensional problems so that the computational cost is significantly reduced the monotonicity and localized linear stability of the variable step scheme are investigated we give some numerical examples to illustrate our results as well as to demonstrate the viability and efficiency of the method over existing methods for the quenching problem it is also shown that the numerical solution obtained preserves important properties of the physical solution of the given problem

moving finite element model for one dimensional infiltration in unsaturated soil the numerical integration of richards equation for infiltration into homogeneous unsaturated soil is carried out by a moving finite element mfe method accurate solution of this one dimensional nonlinear partial differential equation by standard finite difference fd or finite element fe methods can be obtained by using a fine discretization on space and time this can lead to large computer times with the mfe method grid points are moved during computation along the wetting front so that accuracy can be maintained by using a small number of nodes for simulating cases several meters in depth for these cases the mfe method is faster than fd and fe methods the numerical mfe model is verified through comparison with the quasi analytical solution by philip 1957 moreover we compare for some other test problems the computational efficiency of this method with that of two efficient pressure head based fd and fe fixed grid formulations

minimal disks and two convex hypersurfaces in this paper we apply an existence theory and estimates on the morse index of minimal two spheres and minimal disks with free boundary in two convex domains in euclidean space and in manifolds with positive isotropic curvature to study the topology of two convex domains

a general existence theorem for embedded minimal surfaces with free boundary in this paper we prove a general existence theorem for properly embedded minimal surfaces with free boundary in any compact riemannian 3 manifold m with boundary m these minimal surfaces are either disjoint from m or meet m orthogonally the main feature of our result is that there is no assumptions on the curvature of m or convexity of m we prove the boundary regularity of the minimal surfaces at their free boundaries furthermore we define a topological invariant the filling genus for compact 3 manifolds with boundary and show that we can bound the genus of the minimal surface constructed above in terms of the filling genus of the ambient manifold m our proof employs a variant of the min max construction used by colding and de lellis on closed embedded minimal surfaces which were first developed by almgren and pitts 2014 wiley periodicals inc

adaptive local overlapping grid methods for parabolic systems in two space dimensions abstract adaptive mesh refinement techniques are described for two dimensional systems of parabolic partial differential equations solutions are calculated using galerkin s method with a piecewise bilinear basis in space and backward euler integration in time a posteriori estimates of the local discretization error of piecewise bilinear finite element solutions are obtained by a p refinement technique these error estimates are used to control a local h refinement strategy where finer grids are recursively introduced in regions where a prescribed tolerance is exceeded fine grids at a given level of refinement may overlap each other and independent solutions are generated on each of them a version of the schwarz alternating principle is used to coordinate solutions between overlapping fine grids computational results demonstrating the performance of the adaptive procedure on linear and nonlinear problems and apparent convergence of the error estimate for linear heat conduction problem and uniform global refinement is presented

existence of min max free boundary disks realizing the width of a manifold abstract we perform a replacement procedure in order to produce a free boundary minimal surface whose area achieves the min max value over all disk sweepouts of a manifold whose boundary lies in a submanifold our result is based on a proof of the convexity of the energy for free boundary harmonic maps and a generalization of colding minicozzi replacement procedure

a two dimensional mesh moving technique for time dependent partial differential equations abstract we discuss an adaptive mesh moving technique that can be used with a finite difference or finite element scheme to solve initial boundary value problems for systems of partial differential equations in two space dimensions and time the mesh moving technique is based on an algebraic node movement function determined from the geometry and propagation of regions having significant discretization error indicators our procedure is designed to be flexible so that it can be used with many existing finite difference and finite element methods to test the mesh moving algorithm we implemented it in a system code with an initial mesh generator and a maccormack finite difference scheme on quadrilateral cells for hyperbolic vector systems of conservation laws results are presented for several computational examples the moving mesch scheme reduces dispersive errors near shocks and wave fronts and thereby reduces grid requirements necessary to compute accurate solutions while increasing computational efficiency

shock computations with adaptive mesh refinement and moving grids abstract highly accurate shock wave problems are presented using a combination of numerical techniques the new feature of this scheme is the treatment of discontinuities shocks or contact surfaces which are replaced by approximate step functions these approximations permit the solution of the euler equations within the discontinuity and permits a convenient way of treating source terms within a discontinuity the strong conservative law form is used throughout an explicit moving grid technique is used in which a moving frame is found in which the conservation law equations appear stationery in a least squares sense in this moving frame it is possible to drastically reduce or eliminate the temporal truncation errors thus the same time integration scheme can be used throughout the computational domain in the discontinuous as well as continuous regions for accurate physics of wave interactions a nonlinear riemann solver is applied in conjunction with the riemann solver as new waves from such interactions are created an adaptive mesh refinement scheme injects new mesh points automatically to define a post collision wave structure results are presented for the shock tube problem the von neumann spherical blast wave noh s spherical shock problem shock interactions with a set up and step down contact discontinuity

uniqueness and non uniqueness of prescribed mass nls ground states on metric graphs abstract we consider the problem of uniqueness of ground states of prescribed mass for the nonlinear schrodinger energy with power nonlinearity on noncompact metric graphs we first establish that the lagrange multiplier appearing in the nls equation is constant on the set of ground states of mass up to an at most countable set of masses then we apply this result to obtain uniqueness of ground states on two specific noncompact graphs finally we construct a graph that admits at least two ground states with the same mass having different lagrange multipliers our proofs are based on careful variational arguments and rearrangement techniques and hold both for the subcritical range p 2 6 and in the critical case p 6

numerical methods for tracking interfaces an overview is given of special numerical methods for tracking discontinuous fronts and interfaces these methods include surface tracking methods based on connected marker points along the interface volume tracking methods that track the volume occupied by the solution regions bounded by the interfaces and moving mesh methods where the underlying mesh is aligned and moved with the interface the pros and cons of the current methods are discussed and a new method is proposed that overcomes some of the difficulties encountered in approximating equations with multiple interacting interfaces

simple time and space adaptation in one dimensional evolutionary partial differential equations a simple technique for time and space adaptation in one dimensional evolutionary partial differential equations suggested by sanz serna and christie 1 is tested it is found that the equidistribution of hu xi greatly improves on the equidistribution of arc length used by those authors the time step control is found to perform poorly in the integration of rough solutions and the reasons for this behaviour are analysed

stability theory of solitary waves in the presence of symmetry ii abstract consider an abstract hamiltonian system which is invariant under a group of operators we continue to study the effect of the group invariance on the stability of solitary waves applications are given to bound states and traveling wave solutions of nonlinear wave equations

a variable node finite element method abstract a new finite element method with variable nodes for integrating partial differential equations in one space dimension is presented the method utilizes the approximation power of piecewise polynomial functions better than a fixed node finite element method because it distributes grid points in a nonuniform way to suit the peculiarities of the solution at each instant of time the method uses the gradient and curvature of the approximating piecewise polynomial itself to determine the position of the nodes rather than physical properties which vary from problem to problem the method has been implemented in a program vfe which has the exceptional feature of a variable space step as well as a variable time step since the method concentrates nodes in regions where small inter nodal spacing is needed it is very efficient in integrating equations with shock like solutions finally the method can be extended to systems of partial differential equations in one dimension and to equations in two dimensions

threshold phenomena and existence results for nls ground states on metric graphs abstract we investigate the existence of ground states of prescribed mass for the nonlinear schrodinger energy on a noncompact metric graph g while in some cases the topology of g may rule out or on the contrary guarantee the existence of ground states of any given mass in general also some metric properties of g and their quantitative relation with the actual value of the prescribed mass are relevant as concerns the existence of ground states this may give rise to interesting phase transitions from nonexistence to existence of ground states when a certain quantity reaches a critical threshold

review of some adaptive node movement techniques in finite element and finite difference solutions of partial differential equations abstract this article summarizes the results of a literature search for adaptive numerical methods of solving partial differential equations the methods discussed involve the adaptive movement of nodes so as to obtain a low level of solution truncation error while minimizing the number of nodes used in the calculation such methods are applicable to the solution of nonstationary flow problems that contain moving regions of rapid change in the flow variables surrounded by regions of relatively smooth variation flows with shock waves contact surfaces slip streams phase change interfaces and boundary layers can be modelled with great precision by these methods it will be shown that significant economies of execution can be attained if nodes are moved so that they remain concentrated in regions of rapid variation of the flow variables

minimal two spheres and the topology of manifolds with positive curvature on totally isotropic two planes there has been much interest among differential geometers in finding relationships between curvature and topology of riemannian manifolds for the most part efforts have been directed towards understanding the implications of positive or negative sectional curvature ricci curvature or scalar curvature but there are other hypotheses on the curvature which also deserve investigation in this article we will consider the topological implications of a new curvature assumption positive curvature on totally isotropic two planes and we will prove via the sacks uhlenbeck theory of minimal two spheres that a compact simply connected riemannian manifold of dimension at least four with positive curvature on totally isotropic two planes is homeomorphic to a sphere as corollaries we will obtain a proof via minimal two spheres for riemannian manifolds of dimension at least four of the berger klingenberg rauch toponogov sphere theorem strengthened to pointwise quarter pinching as well as a proof that every simply connected compact riemannian manifold with positive curvature operators is homeomorphic to a sphere let m be an n dimensional riemannian manifold with tangent space tpm at the point p e m recall that the curvature operator at p is the self adjoint

grid optimization for the boundary element method the quality of solution obtained using the boundary element method bem is dependent on how the boundary is discretized this is particularly true in domains of complex geometry a rule for grid optimization for the bem is derived on the bases of an asymptotic measure of the boundary element error that preserves the number of elements degrees of freedom three example problems are provided to show the advantages of grid optimization in terms of accuracy and cost

a simplified adaptive mesh technique derived from the moving finite element method an adaptive mesh technique is derived as a constrained minimization of the functional used in the moving finite element mfe method the result is a matrix equation for mesh velocities alone i e a pure adaptive mesh prescription the method retains many attractive features of mfe including the ability to independently control mesh motion through penalty terms

sharp trace hardy sobolev maz ya inequalities and the fractional laplacian in this work we establish trace hardy and trace hardy sobolev maz ya inequalities with best hardy constants for domains satisfying suitable geometric assumptions such as mean convexity or convexity we then use them to produce fractional hardy sobolev maz ya inequalities with best hardy constants for various fractional laplacians in the case where the domain is the half space our results cover the full range of the exponent s in 0 1 of the fractional laplacians in particular we give a complete answer in the l2 setting to an open problem raised by frank and seiringer sharp fractional hardy inequalities in half spaces in around the research of vladimir maz ya international mathematical series 2010

an adaptive method of lines with error control for parabolic equations of the reaction diffusion type abstract a piecewise linear finite element based method of lines is presented for the numerical solution of coupled parabolic partial differential equations which model biological and physicochemical reaction diffusion processes in one space dimension the vertial lines emanating from the space nodes in this method change at automatically selected times when in order to control a norm of the space discretization error adaptive spatial regridding occurs the regridding algorithm is an extension of one described previously by the authors 7 and is implemented in the program femol 1 which uses the lsodi package 14 15 of hindmarsh and painter to integrate the ordinary differential equations in time along the vertical lines computational results show that the method is efficient that a posteriori estimates of the space discretization error are accurate and that the adaptive procedure reliably controls the space discretization error

moving finite element methods for evolutionary problems ii applications abstract in this the second of two papers on the subject we present applications of the moving finite element method to a number of test problems key features are linear elements a direct approach to parallelism and node overtaking avoiding penalty functions rapid inversion of the mass matrix by preconditioned conjugate gradients and explicit euler time stepping the resulting codes are fast and efficient and are able to follow fronts and similar features with great accuracy the paper includes a substantial section on changes of dependent variable and front tracking techniques for non linear diffusion problems test problems include non linear hyperbolic conservation laws and non linear parabolic equations in one and two dimensions

min max theory and the willmore conjecture in 1965 t j willmore conjectured that the integral of the square of the mean curvature of a torus immersed in r 3 is at least 2 2 we prove this conjecture using the min max theory of minimal surfaces

iterative schemes and algorithms for adaptive grid generation in one dimension some iterative adaptive grid generators developed by the author are numerically explored in detail to assess their relative merits against conventional grid generators based on a direct method of integration and interpolation we find that some of these iterative adaptive grid generators are preferable to a direct method of integration and interpolation in contrast with a direct method appropriate use of these iterative adaptive grid generators produces adaptive grids with a smooth variation of the grid spacing ratio and resolution all adaptive grid generators are a subclass of a more general iterative map general features of this iterated map which are related to the function to be resolved are briefly discussed the results obtained here supplements recent investigations on these adaptive grid generators by the author

the finite element method for parabolic equations ii a posteriori error estimation and adaptive approach abstract we extend in this paper the analysis of a posteriori estimates of the space discretization error presented in a previous paper 3 for time independent space meshes in the context of the model problem studied there results are given relating the effectivity of the error estimator to properties of the solution space meshes and manner in which the meshes change a procedure based upon this theory is presented for the adaptive construction of time dependent meshes the results of some computational experiments show that this procedure is practically very effective and suggest that it can be used to control the space discretization error in more general problems author

adaptive mesh techniques for fronts in star formation abstract we present an implicit adaptive grid finite difference technique specifically designed to locate and track arbitrary fronts interfaces and structures in a radiation hydro flow the adaptive mesh is constructed in such a way that the average change of the flow variables per grid zone is the same throughout the entire grid a local grid refinement of up to a factor of 10 6 compared to an equidistant mesh has been obtained on a machine with 14 decimal digits per word crucial to this type of approach is the use of a stiffness operator which prevents tangling of the mesh the elliptic equation for the grid motion is solved implicitly coupled with the equations for the flow variables per timestep very steep and narrow flow features can travel many times their own width in true space however with respect to the adaptive mesh the motion of such flow features is always less than one zone thus the courant friedrichs lewy 1 cfl condition s accuracy limitation is always fulfilled although its stability criterion does not pose any timestep limitations for our implicit numerical method in fact the choice of timesteps is based purely on physical and accuracy considerations typical applications run on courant numbers ranging from 0 1 to 10 12 the performance of this numerical scheme is demonstrated on a planar 1 d shocktube and the formation of a one solar mass protostar in spherical symmetry

unified approach to simulation on deforming elements with application to phase change problems abstract several different approaches to simulation on deforming finite elements are shown to generate the same weighted residuals formulation for the evolution of the dependent variables control of node motion by means of mesh stretching in two dimensions is illustrated in the context of phase change problems simulation of problems with large phase boundary excursions shows good agreement with analytic solutions

parallel lod methods for second order time dependent pdes abstract time dependent problems in partial differential equations pdes are often solved by the method of lines mol in many important instances the exact solution of the resulting system of ordinary differential equations odes satisfies recurrence relations involving the matrix exponential function this function is approximated by a new type of rational function possessing real and distinct poles which consequently readily admits a partial fraction expansion thereby allowing the distribution of the work in solving the corresponding linear algebraic systems in essentially backward euler like solves on concurrent processors locally one dimensional lod splitting methods with enhanced parallelization are developed for multispace problems utilizing strang like splitting techniques the use of rational approximants with distinct real poles in the temporal direction in collusion with splitting techniques in the spatial directions creates the potential for efficient coarse grain time stepping parallel algorithms on mimd machines the resulting parallel algorithms possess appropriate stability properties and are implemented on various parabolic and hyperbolic pdes from the literature in higher space dimensional problems

the use of nonrational functions to represent steep front solutions to partial differential equations it is shown by calculated examples that it is possible to follow successfully the development and evolution of steep fronts almost to but not including actual discontinuities in the solution of time dependent partial differential equations by means of global non rational representations a characteristic steplike shape forms the basis for the representations the hyperbolic tangent and a function involving a square root both having this characteristic shape gave similar results the method of lines specifically as formulated by gelinas doss and miller j comput phys 40 1981 202 249 was used to form a system of ordinary differential equations governing the time variation of the parameters in the representation a fixed shape moving solution of the wave equation and developing steep fronts in solutions of burgers equation and of the buckley leverett equation were successfully computed the results are all displayed in the form of plotted profiles and are compared with the corresponding results in the literature

numerical quenching for the semilinear heat equation with a singular absorption in this paper we study the numerical approximation for the heat equation with a singular absorption we prove that the numerical quenching rate coincides with the continuous one we also see that the quenching time and the quenching set converge to the continuous one in fact under some restriction on the initial data the numerical quenching coincides with the continuous one finally we give some numerical results to illustrate our analysis

some aspects of adaptive grid technology related to boundary and interior layers we consider the use of adaptive mesh strategies for solution of problems exhibiting boundary and interior layer solutions as the presence of these layer structures suggests reliable and accurate solution of this class of problems using finite difference finite volume or finite element schemes requires grading the mesh into the layers and due attention to the associated algorithms when the nature and structure of the layer is known mesh grading can be achieved during the grid generation by specifying an appropriate grading function however in many applications the location and nature of the layer behavior is not known in advance consequently adaptive mesh techniques that employ feedback from intermediate grid solutions are an appealing approach in this paper we provide a brief overview of the main adaptive grid strategies in the context of problems with layers associated error indicators that guide the refinement feedback control grid optimization process are also covered and there is a brief commentary on the supporting data structure requirements some current issues concerning the use of stabilization in conjunction with adaptive mesh refinement amr the question of pollution effects in computation of local error indicators the influence of nonlinearities and the design of meshes for targeted optimization of specific quantities are considered the application of amr for layer problems is illustrated by means of case studies from semiconductor device transport drift diffusion nonlinear reaction diffusion layers due to surface capillary effects and shockwaves in compressible gas dynamics

bubble tree convergence for harmonic maps let be a compact riemann surface any sequence fn m of harmonic maps with bounded energy has a bubble tree limit consisting of a harmonic map o m and a tree of bubbles fk s2 m we give a precise construction of this bubble tree and show that the limit preserves energy and homotopy class and that the images of the fn converge pointwise we then give explicit counterexamples showing that bubble tree convergence fails i for harmonic maps fn when the conformal structure of varies with n and ii when the conformal structure is fixed and n is a palais smale sequence for the harmonic map energy

the moving finite element method applications to general partial differential equations with multiple large gradients abstract the moving finite element mfe method has been reduced to practice in the automatic solution program dyla for general systems of transient partial differential equations pdes in 1 d several test examples are presented which illustrate the unique node movement and systematic control features which are intrinsic in the mfe method computational dilemmas of numerical diffusion gibbs overshooting and undershooting zone tangling and grid remap or re connection aliasing which occur frequently in conventional pde methods are essentially eliminated in the mfe method arbitrarily large gradients or shocks can be solved with extremely high resolution and accuracy for non coincident or even counterdirected propagating wavefronts boundary layers of arbitrarily small dimensions are solved with high accuracy simultaneously with the large scale structures in reactive and non reactive fluid calculations the mfe method requires a small fraction of the grid nodes which are used in conventional pde solution methods because the nodes migrate continuously and systematically to those positions where they are most needed in order to yield accurate pde solutions on entire problem domains courant friedrichs lewy time step limits are exceeded by wide margins by factors of two to several thousand finally the extension of the mfe method to 2 d is briefly discussed

a new moving finite element method based on quadratic approximation functions moving finite element methods are adaptive gridding procedures especially designed for systems of partial differential equations whose solutions contain steep gradients a new moving finite element method based on quadratic approximation functions is presented both the theoretical and computational aspects are outlined performance of the method is illustrated with solutions to burgers equation the solution is accurate and remarkably smooth in the entire domain

a simple adaptive technique for nonlinear wave problems abstract a method for the numerical integration of the nonlinear schrodinger equation is derived which uses variable time steps and a moving spatial grid the benefits of adaptation are clearly demonstrated in the numerical experiments reported the simple technique employed to move the nodes can be applied with little coding effort to general one dimensional systems of pdes

the finite element method for parabolic equations in this first of two papers computable a posteriori estimates of the space discretization error in the finite element method of lines solution of parabolic equations are analyzed for time independent space meshes the effectiveness of the error estimator is related to conditions on the solution regularity mesh family type and asymptotic range for the mesh size for clarity the results are limited to a model problem in which piecewise linear elements in one space dimension are used the results extend straight forwardly to systems of equations and higher order elements in one space dimension while the higher dimensional case requires additional considerations the theory presented here provides the basis for the analysis and adaptive construction of time dependent space meshes which is the subject of the second paper computational results show that the approach is practically very effective and suggest that it can be used for solving more general problems

boundary regularity and the dirichlet problem for harmonic maps in a previous paper 10 we developed an interior regularity theory for energy minimizing harmonic maps into riemannian manifolds in the first two sections of this paper we prove boundary regularity for energy minimizing maps with prescribed dirichlet boundary condition we show that such maps are regular in a full neighborhood of the boundary assuming appropriate regularity on the manifolds the boundary and the data the reader may refer to theorem 2 7 for a statement of the precise result it is not surprising that the boundary regularity is actually stronger than the partial regularity we obtained for the interior this is due to the fact that there are no nontrivial smooth harmonic maps from hemispheres s which map the boundary s 9s to a point i j n 2 and is analogous to the fact that in certain cases we were able to obtain complete regularity in the interior many authors have worked on boundary regularity for this general type of problem we mention hildebrandt and widman 5 and hamilton 4 as having obtained important results specifically for harmonic maps morrey had obtained the boundary regularity for domain dimension n 2 in conjunction with his investigation of the plateau problem in riemannian manifolds 8 in 3 of this paper we observe that the direct method gives solvability of the dirichlet problem under reasonable hypotheses on the manifolds we give as an application an amusing proof of a theorem of sacks and uhlenbeck 9 on the existence of minimal 2 spheres representing the second homotopy group of a manifold the same method gives smooth harmonic representations for k n for a certain class of manifolds n these are characterized by the nonexistence of lower dimensional harmonic spheres whose homogeneous extensions are minimal see proposition 3 4 in the last section of the paper we discuss approximation of l maps by smooth maps we give a simple example of an l map from the three dimensional ball to the two sphere which is not an l limit of continuous maps we