time dependent ginzburg landau equations of superconductivity abstract we study in this article the existence uniqueness and long time behavior of the solutions of a nonstationary ginzburg landau superconductivity model we first prove the existence and uniqueness of solutions with h 1 initial data which are crucial for the study of the global attractor we also obtain for the first time the existence of global weak solutions of the model with l 2 initial data it is then proved that the ginzburg landau system admits a global attractor which represents exactly all the long time dynamics of the system the global attractor obtained consists of exactly the set of steady state solutions and its unstable manifold its hausdorff and fractal dimensions are estimated in terms of the physically relevant ginzburg landau parameter diffusion parameter and applied magnetic field we construct explicitly absorbing sets for some abstract semigroups having a lyapunov functional and consequently prove the existence of global attractors this abstract result is applied to the ginzburg landau system for which the existence of global attractor does not seem to be the direct consequence of some a priori estimates of solutions

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

quasilinear elliptic system arising in a three dimensional type ii superconductor for infinite κ abstract we study a quasilinear elliptic system arising in a three dimensional superconductor this model is formally derived from the ginzburg landau energy at for a meissner solution if the tangential trace of the magnetic field h is given on we prove the existence of a unique solution for small data and nonexistence for large data on the other hand we prove that the current j curl h is such that j 2 is maximum on the boundary

approximations of a ginzburg landau model for superconducting hollow spheres based on spherical centroidal voronoi tessellations in this paper the numerical approximations of the ginzburg landau model for a superconducting hollow spheres are constructed using a gauge invariant discretization on spherical centroidal voronoi tessellations a reduced model equation is used on the surface of the sphere which is valid in the thin spherical shell limit we present the numerical algorithms and their theoretical convergence as well as interesting numerical results on the vortex configurations properties of the spherical centroidal voronoi tessellations are also utilized to provide a high resolution scheme for computing the supercurrent and the induced magnetic field

symmetric vortices for the ginzberg landau equations of superconductivity and the nonlinear desingularization phenomenon abstract the existence of countably many distinct symmetric vortices for the nonlinear ginzberg landau equations of superconductivity with an arbitrary positive parameter is proven we then prove the existence of a linearization phenomenon as the parameter in this case the linearized equations contain an appropriate dirac delta function these equations are known as the london equations

convergence analysis of a numerical method for a mean field model of superconducting vortices in the mean field models of superconductivity the individual vortex like structures occurring in practical type ii superconductors are averaged and a vortex density is solved for them the numerical solution of the mean field models makes large scale simulations of vortex phenomena possible in this paper we present a simple convergence analysis for a numerical method based on hybrid finite element finite volume finite difference approximations in the two dimensional case by providing various different interpretations to the discretization scheme

on the zero set of the wave function in superconductivity we consider the ginzburg landau functional in a multiply connected planar domain with enclosed magnetic flux particular attention is given to the zero set of the order parameter we show that there exist applied fields for which the zero set is of codimension 1

superconductivity near the normal state in a half plane under the action of a perpendicular electric current and an induced magnetic field we consider the linearization of the time dependent ginzburg landau system near the normal state we assume that a constant magnetic field and an electric current are applied through the sample which captures half of the plane inducing thereby a linearly varying magnetic field in the limit of small normal conductivity we prove that if the electric current is lower than some critical value the normal state loses its stability for currents stronger than this critical value the normal state is stable to obtain this stability result we analyze both the spectrum and the pseudo spectrum of the linearized operator the critical current tends in this small conductivity limit to another critical current which had been obtained for a reduced model which neglects magnetic field effects

mixed normal superconducting states in the presence of strong electric currents we study the ginzburg landau equations in the presence of large electric currents that are smaller than the critical current where the normal state loses its stability for steady state solutions in the large kappa limit we prove that the superconductivity order parameter is exponentially small in a significant part of the domain and simply small in the rest of it similar results are obtained for the time dependent problem in continuation of the paper by the two first authors 3 we conclude by obtaining some weaker results albeit similar for steady state solutions in the large domain limit

spectral methods for nonlinear parabolic systems many physical problems are naturally formulated as nonlinear parabolic systems of partial differential equations in periodic geometry in this paper a simple efficient spectrally accurate numerical method for these problems is described and implemented the method combines stiff extrapolation with fast solvers for elliptic systems theory and numerical results show that the method solves even difficult problems including phase field models and mean curvature flows

stability and convergence of finite element approximation schemes for harmonic maps this article discusses stability and convergence of approximation schemes for harmonic maps a finite element discretization of an iterative algorithm due to f alouges is introduced and shown to be stable and convergent in general only on acute type triangulations an break a posteriori criterion is proposed which allows us to monitor sufficient conditions for weak convergence to a harmonic map on general triangulations and for adaptive mesh refinement numerical break experiments show that an adaptive strategy automatically refines triangulations in neighborhoods of typical point singularities and thereby underline its efficiency

a new approach for numerical simulation of the time dependent ginzburg landau equations we introduce a new approach for finite element simulations of the time dependent ginzburg landau equations tdgl in a general curved polygon possibly with reentrant corners specifically we reformulate the tdgl into an equivalent system of equations by decomposing the magnetic potential to the sum of its divergence free and curl free parts respectively numerical simulations of vortex dynamics show that in a domain with reentrant corners the new approach is much more stable and accurate than the traditional approaches of solving the tdgl directly under either the temporal gauge or the lorentz gauge in a convex domain the new approach gives comparably accurate solutions as the traditional approaches

adaptive galerkin methods with error control for a dynamical ginzburg landau model in superconductivity the time dependent ginzburg landau model which describes the phase transitions taking place in superconductors is a coupled system of nonlinear parabolic equations it is discretized semi implicitly in time and in space via continuous piecewise linear finite elements a posteriori error estimates are derived for the l infty l 2 norm by studying a dual problem of the linearization of the original system other than the dual of error equations numerical simulations are included which illustrate the reliability of the estimators and the flexibility of the proposed adaptive method

an alternating crank nicolson method for decoupling the ginzburg landau equations the time dependent ginzburg landau model of superconductors consists of coupled nonlinear partial differential equations which presents difficulties in the numerical solution we present an alternating crank nicolson method for this model that leads to two decoupled algebraic subsystems one is linear and the other is semilinear both have nice numerical properties and can be solved by efficient matrix solvers we show the stability and convergence and derive error estimates for this scheme numerical results are also reported

local minimizers with vortex pinning to ginzburg landau functional in three dimensions abstract we consider the pinning effect for full ginzburg landau functional the existence of local minimizers with vortices locating in the pinning regions is obtained

nucleation of instability of the meissner state of 3 dimensional superconductors this paper concerns a nonlinear partial differential system in a 3 dimensional domain involving the operator curl2 which is a simplified model used to examine nucleation of instability of the meissner state of a superconductor as the applied magnetic field reaches the superheating field we derive a priori c 2 estimates for a weak solution h the curl of the magnetic potential and determine the location of the maximal points of curlh which correspond to the nucleation of instability of the meissner state we show that if the penetration length is small the solution exhibits a boundary layer if the applied magnetic field is homogeneous curlh is maximal around the points on the boundary where the applied field is tangential to the surface

vortex filaments pinning for inhomogeneous ginzburg landau equations abstract in this paper the structure of vortices of the ginzburg landau system with impurity in higher dimensions was studied this functional has been proposed as a model for vortex pinning we show that vortices are localized near the set of minimum of impurity this is the pinning phenomenon predicted in the physics literature it was also proven that the limiting vortices is n 2 dimensional rectifiable set

application of differential transformation to eigenvalue problems using differential transformation to solve eigenvalue problems is introduced in this study first differential transformation is introduced briefly second taking differential transformation of the strum liouville problem a set of difference equations is derived doing some simple mathematics operations on these operations we can get any ith eigenvalue and eigenfunction at a time finally two eigenvalue problems are solved by the present method the calculated results are compared closely with the results obtained by another analytical method

surface superconductivity in 3 dimensions we study the ginzburg landau system for a superconductor occupying a 3 dimensional bounded domain and improve the estimate of the upper critical field h c3 obtained by k lu and x pan in j diff eqns 168 2000 386 452 we also analyze the behavior of the order parameters we show that under an applied magnetic field lying below and not far from h c3 order parameters concentrate in a vicinity of a sheath of the surface that is tangential to the applied field and exponentially decay both in the normal and tangential directions away from the sheath in the l 2 sense as the applied field decreases further but keeps in between and away from h c2 and h c3 the superconducting sheath expands but does not cover the entire surface and superconductivity at the surface portion orthogonal to the applied field is always very weak this phenomenon is significantly different to the surface superconductivity on a cylinder of infinite height studied by x pan in comm math phys 228 2002 327 370 where under an axial applied field lying in between h c2 and h c3 the entire surface is in the superconducting state

numerical methods for simulating ginzburg landau vortices numerical solution to the ginzburg landau gl equation becomes infeasible as the gl parameter kappa and the number of gl vortices increase to a physically interesting regime it is in this regime that we focus our attention to design a simulated annealing approach for minimizing the associated renormalized energy to study the gl vortex configurations this approach applied to a commonly used model produces vortex configurations which are of interest in recent studies

a posteriori error analysis for time dependent ginzburg landau type equations this work presents an a posteriori error analysis for the finite element approximation of time dependent ginzburg landau type equations in two and three space dimensions the solution of an elliptic self adjoint eigenvalue problem as a post processing procedure in each time step of a finite element simulation leads to a fully computable upper bound for the error theoretical results for the stability of degree one vortices in ginzburg landau equations and of generic interfaces in allen cahn equations indicate that the error estimate only depends on the inverse of a small parameter in a low order polynomial the actual dependence of the error estimate upon this parameter is explicitly determined by the computed eigenvalues and can therefore be monitored within an approximation scheme the error bound allows for the introduction of local refinement indicators which may be used for adaptive mesh and time step size refinement and coarsening numerical experiments underline the reliability of this approach

on stability of abrikosov vortex lattices abstract the ginzburg landau equations play a key role in superconductivity and particle physics they inspired many imitations in other areas of physics these equations have two remarkable classes of solutions vortices and abrikosov vortex lattices for the standard cylindrical geometry the existence theory for these solutions as well as the stability theory of vortices are well developed the latter is done within the context of the time dependent ginzburg landau equations the gorkov eliashberg schmid equations of superconductivity and the abelian higgs model of particle physics we study stability of abrikosov vortex lattices under finite energy perturbations satisfying a natural parity condition both defined precisely in the text for the dynamics given by the gorkov eliashberg schmid equations for magnetic fields close to the second critical magnetic field and for arbitrary lattice shapes we prove that there exist two functions on the space of lattices such that abrikosov vortex lattice solutions are asymptotically stable provided the superconductor is of type ii and these functions are positive and unstable for superconductors of type i or if one of these functions is negative

vibrations of non homogeneous rectangular membranes abstract an exact solution for the title problem is obtained for the case where the two subdomains of the non homogeneous membrane are of rectangular shape it is shown that the fundamental frequency coefficient obtained by means a of the classical kantorovish method and b the optimized kantorovich method are in very good agreement with the result predicted by the exact method

new types of vibration modes for stepped membranes frequency spectra for vibrations of rectangular membranes with a step discontinuity in density along a line parallel to an edge are investigated particularly with a view to finding explicit analytical solutions cases of exact harmonic sub spectra with explicit trigonometrical mode functions are found it is stressed that the complete set of modes also includes hyperbolic functions in the description of the amplitude functions for one section of the membrane furthermore modes with a straight line segment in cross section are exposed at the transition between trigonometric and hyperbolic amplitude functions for certain critical density ratio values tables of spectra which incorporate numerically determined frequencies are presented for several configurational examples features such as mode crossing are revealed

numerical study of quantized vortex interaction in complex ginzburg landau equation on bounded domains in this paper we study numerically quantized vortex dynamics and their interaction in the two dimensional complex ginzburg landau equation cgle with a dimensionless parameter e 0 on bounded domains under either dirichlet or homogeneous neumann boundary condition we begin with a review of the reduced dynamical laws rdls for time evolution of quantized vortex centers in cgle and show how to solve these nonlinear ordinary differential equations numerically then we present efficient and accurate numerical methods for solving the cgle on either a rectangular or a disk domain under either dirichlet or homogeneous neumann boundary condition based on these efficient and accurate numerical methods for cgle and the rdls we explore rich and complicated quantized vortex dynamics and interaction of cgle with different e and under different initial physical setups including single vortex vortex pair vortex dipole and vortex lattice compare them with those obtained from the corresponding rdls and identify the cases where the rdls agree qualitatively and or quantitatively as well as fail to agree with those from cgle on vortex interaction finally we also obtain numerically different patterns of the steady states for quantized vortex lattices in the cgle dynamics on bounded domains

numerical solution of the three dimensional ginzburg landau models using artificial boundary for three dimensional simulations of the vortex phenomena in superconductors based on the ginzburg landau gl theory the physical variables must be solved in the whole space in general exact boundary conditions on an artificial boundary are discussed to reformulate the problem in a finite domain approximations of these boundary conditions with high accuracy are given and their convergence properties are examined error estimates are derived for the finite element approximations with the approximate boundary conditions

analysis and approximation of the ginzburg landau model of superconductivity the authors consider the ginzburg landau model for superconductivity first some well known features of superconducting materials are reviewed and then various results concerning the model the resultant differential equations and their solution on bounded domains are derived then finite element approximations of the solutions of the ginzburg landau equations are considered and error estimates of optimal order are derived

a model for superconducting thin films having variable thickness abstract a two dimensional macroscopic model for superconductivity in thin films having variable thickness is derived through an averaging process across the film thickness the resulting model is similar to the well known ginzburg landau equations for homogeneous isotropic materials except that a function that describes the variations in the thickness of the film now appears in the coefficients of the differential equations some results about solutions of the variable thickness model are then given including existence of solutions and boundedness of the order parameter it is also shown that the model is consistent in the sense that solutions obtained from the new model are an appropriate limit of a sequence of averages of solutions of the three dimensional ginzburg landau model as the thickness of the film tends to zero an application of the variable thickness thin film model to flux pinning is then provided in particular the results of numerical calculations are given that show that the vortex like structures that are present in certain superconductors are attracted re relatively thin regions in a material sample finally extensions of the model to other settings are discussed

sweeping algorithms for inverting the discrete ginzburg landau operator the ginzburg landau equations we study arise in the modeling of superconductivity one widely used method of discretizing the equations together with the associated periodic boundary conditions in the case of a rectangle of dimension 2 leads to a five point stencil solving the system means inverting a sparse matrix of dimension n 2 where n is the number of grid points on each side of the rectangle we propose a method that is similar to the shooting technique in the numerical solution of ordinary differential equations for small n the method requires inverting a full matrix of dimension 2n when n is large an iterative procedure combining partial sweeping and the technique of divide and conquer domain decomposition is appropriate

on the transition to the normal phase for superconductors surrounded by normal conductors abstract for a cylindrical superconductor surrounded by a normal material we discuss transition to the normal phase of stable locally stable and critical configurations associated with those phase transitions we define critical magnetic fields and we provide a sufficient condition for which those critical fields coincide in particular when the conductivity ratio of the superconducting and the normal material is large we show that the aforementioned critical magnetic fields coincide thereby proving that the transition to the normal phase is sharp one key ingredient in the paper is the analysis of an elliptic boundary value problem involving transmission boundary conditions another key ingredient involves a monotonicity result with respect to the magnetic field strength of the first eigenvalue of a magnetic schrodinger operator with discontinuous coefficients

implicit integration of the time dependent ginzburg landau equations of superconductivity this article is concerned with the integration of the time dependent ginzburg landau tdgl equations of superconductivity four algorithms ranging from fully explicit to nonlinearly implicit are presented and evaluated for stability accuracy and compute time the benchmark problem for the evaluation is the equilibration of a vortex configuration in a superconductor that is embedded in a thin insulator and subject to an applied magnetic field

transverse vibrations of composite membranes of arbitrary boundary shape this paper deals with an approximate method based on the technique of conformal mapping and a variational formulation which allows for a straightforward solution of the title problem for a certain type of composite membranes numerical results are presented for membranes of regular polygonal shape whose inner circular core possesses a density 1 while the remaining is characterized by 2 the fundamental eigenvalues are also verified by the fourier expansion collocation method and by a finite element algorithm

finite element methods for the time dependent ginzburg landau model of superconductivity abstract the initial boundary value problem for the time dependent ginzburg landau equations that model the macroscopic behavior of superconductors is considered the convergence of finite dimensional semidiscrete galerkin approximations is studied as is a fully discrete scheme the results of some computational experiments are presented

on the discretization error of parametrized nonlinear equations many applications lead to nonlinear parameter dependent equations h y t y 0 where h y times t to y y 0 in operatorname rge h and the state space y is infinite dimensional while the parameter space t has finite dimension the case dim t 1 is of special interest in connection with continuation methods for this case a general theory is developed which provides for the existence of solution paths of a rather general class of such equations and of their finite dimensional approximations and which allows for an assessment of the error between these paths a principal tool in this analysis is the theory of nonlinear fredholm operators the results cover a more general class of operators than the mildly nonlinear mappings to which other approaches appear to be restricted

homotopy classification of minimizers of the ginzburg landau energy and the existence of permanent currents our objective is to explain the phenomenon of permanent currents within the context of the ginzburg landau model for superconductors using variational techniques we make a connection between the formation of permanent currents and the topology of the superconducting sample

bifurcation analysis for phase transitions in superconducting rings with nonuniform thickness we perform an analytic study of the problem of transitions between normal and superconducting phases for a sample which encloses a magnetic flux as in the classic little parks experiment for a sample of uniform thickness the order parameter is uniform but even infinitesimal deviations from uniform thickness give rise to a mixed state which avoids enclosing the magnetic field the stability domain of this mixed state is a line segment in the magnetic field temperature plane delimited by two critical points the phase diagram contains several bifurcation lines which are systematically analyzed

unconditional superconvergence analysis of a linearized crank nicolson galerkin fem for generalized ginzburg landau equation abstract in this paper we study an effective numerical method for the generalized ginzburg landau equation gle based on the linearized crank nicolson difference method in time and the standard galerkin finite element method with bilinear element in space the time discrete and space time discrete systems are both constructed we focus on a rigorous analysis and consideration of unconditional superconvergence error estimates of the discrete schemes firstly by virtue of the temporal error results the regularity for the time discrete system is presented secondly the classical ritz projection is used to obtain the spatial error with order o h 2 in the sense of l 2 norm thanks to the relationship between the ritz projection and the interpolated projection the superclose estimate with order o 2 h 2 in the sense of h 1 norm is derived thirdly it follows from the interpolated postprocessing technique that the global superconvergence result is deduced finally some numerical results are provided to confirm the theoretical analysis

spinor ginzburg landau equation and mean curvature flow abstract recent papers in the physics literature have introduced spin coupled or spinor ginzburg landau models for complex vector valued order parameters in order to account for ferromagnetic or antiferromagnetic effects in high temperature superconductors in this paper we study the asymptotic behavior of solutions of the evolutionary spinor ginzburg landau equation in three dimensions we prove also that asymptotically the singularities evolve according to mean curvature flow in the sense of weak formulation

numerical approximations of the ginzburg landau models for superconductivity in this paper we review various methods for the numerical approximations of the ginzburg landau models of superconductivity particular attention is given to the different treatment of gauge invariance in both the finite element finite difference and finite volume settings representative theoretical results typical numerical simulations and computational challenges are presented generalizations to other relevant models are also discussed

a higher order asymptotic approximation for the fundamental frequency of a multiply connected membrane abstract the fundamental frequency of a fixed membrane is the square root of the lowest eigenvalue of negative laplace operator with dirichlet boundary conditions a multiply connected membrane with inner cores of vanishing maximal dimensions 2 c j is considered in the present article the modified perturbation method developed for a doubly connected membrane is extended to provide a general formula for the fundamental frequency of the multiply connected membrane a higher order asymptotic approximation as c j 0 for the fundamental frequency of a membrane with inner circular cores of radius c j is specified it is an excellent extension of the results in the literature moreover a second order asymptotic approximation as c 0 for the fundamental frequency of a circular membrane of radius 1 with finitely many inner circular cores of small radius c is found and computed explicitly the effects of the positions of the inner cores on the second order asymptotic approximation are investigated the accuracy of the second order asymptotic approximation is also shown by the comparisons among the asymptotic approximations and the numerical values computed by other investigators

comments on magnetic line source diffraction by an impedance step our recent research has revealed several errors in 1 eq 20 these were identified through comparisons with 2 eq 26 a negative sign is missing in the expression in the second term furthermore the missing factors and are to be placed in the denominator of the second and the third terms respectively also there is another error in the denominator of the second term that is which is to be correctly written as the modifications and suggestions presented in the comment make the obtained results in 1 more meaningful and valid

mixed finite element methods for a dynamical ginzburg landau model in superconductivity in this paper we consider the numerical solutions of the nonlinear time dependent ginzburg landau model which describes the phase transitions taking place in superconducting films we propose a semi implicit finite element scheme which is based on a linear finite element approximation of the order parameter psi and a mixed finite element discretization for the equation involving the magnetic potential a the error estimates of the scheme are derived

sobolev gradients and the ginzburg landau functional we describe a sobolev gradient method for finding minima of the ginzburg landau functional for superconductivity this method leads to a particularly simple algorithm which avoids consideration of the nonlinear boundary conditions associated with the ginzburg landau equations

remark on a relation between magnetic field and topological degree of order parameter in a type ii superconductivity model abstract it is proved that in the absence of an applied magnetic field the two space dimensional steady state helmholtz free energy which models the type ii superconductors admits no minimizer in the following admissible function set u a h 1 c h 1 r 2 u 1 1 2 i u d u ud u d 2k here c is the complex plane k is a fixed nonzero integer which is the topological degree of u with respect to its value 0 the case k 0 is discussed separately under different assumptions

asymptotic analysis of a secondary bifurcation of the one dimensional ginzburg landau equations of superconductivity the bifurcation of asymmetric superconducting solutions from the normal solution is considered for the one dimensional ginzburg landau equations by the methods of formal asymptotics the behavior of the bifurcating branch depends on the parameters d the size of the superconducting slab and kappa the ginzburg landau parameter the secondary bifurcation in which the asymmetric solution branches reconnect with the symmetric solution branch is studied for values of kappa d for which it is close to the primary bifurcation from the normal state these values of kappa d form a curve in the kappa d plane which is determined at one point on this curve called the quintuple point the primary bifurcations switch from being subcritical to supercritical requiring a separate analysis the results answer some of the conjectures of a aftalion and w c troy phys d 132 1999 pp 214 232

a new compact finite difference scheme for solving the complex ginzburg landau equation the complex ginzburg landau equation is often encountered in physics and engineering applications such as nonlinear transmission lines solitons and superconductivity however it remains a challenge to develop simple stable and accurate finite difference schemes for solving the equation because of the nonlinear term most of the existing schemes are obtained based on the crank nicolson method which is fully implicit and must be solved iteratively for each time step in this article we present a fourth order accurate iterative scheme which leads to a tri diagonal linear system in 1d cases we prove that the present scheme is unconditionally stable the scheme is then extended to 2d cases numerical errors and convergence rates of the solutions are tested by several examples

the fundamental frequency of non homogeneous rectangular membranes the present study deals with the case in which the density of the membrane is a linear function of the x variable two independent solutions are obtained using the optimized galerkin kantorovich approach and the differential quadrature method

linearized adi schemes for two dimensional space fractional nonlinear ginzburg landau equation abstract space and time approximations for two dimensional space fractional complex ginzburg landau equation are examined the schemes under consideration are discreted by the second order backward differential formula bdf2 in time and two classes of the fractional centered finite difference methods in space a linearized technique is employed by the extrapolation we prove the unique solvability and stability for both numerical methods the convergence of both numerical methods is analyzed at length utilizing the energy argument and the convergence orders under the optimal step size ratio are o 2 h 2 and o 2 h 4 in the sense of the discrete l 2 norm where is the time step size h max h x h y and h x h y are spatial grid sizes in the x direction and y direction respectively in addition we construct a multistep alternating direction implicit adi scheme and a multistep compact adi scheme based on bdf2 for the efficiently numerical implementation finally numerical examples are carried out to verify our theoretical results

optimal error estimates of linearized crank nicolson galerkin fems for the time dependent ginzburg landau equations in superconductivity in this paper we study linearized crank nicolson galerkin finite element methods for time dependent ginzburg landau equations under the lorentz gauge we present an optimal error estimate for the linearized schemes almost unconditionally i e when the spatial mesh size h and the temporal step tau are smaller than a given constant while previous analyses were given only for some schemes with strong restrictions on the time step size the key to our analysis is the boundedness of the numerical solution in some strong norm we prove the boundedness for the cases tau ge h and tau le h respectively the former is obtained by a simple inequality with which the error functions at a given time level are bounded in terms of their average at two consecutive time levels and the latter follows a traditional way with the induction inverse inequality two numerical examples are investigated to confirm our theoretical analysis and to show clearly that no time step condition is needed

finite dimensional approximation of nonlinear problems in the first two papers of this series 4 5 we have studied a general method of approximation of nonsingular solutions and simple limit points of nonlinear equations in a banach space we derive here general approximation results of the branches of solutions in the neighborhood of a simple bifurcation point the abstract theory is applied to the galerkin approximation of nonlinear variational problems and to a mixed finite element approximation of the von karman equations

finite element approximation of a periodic ginzburg landau model for type ii superconductors we consider efficient finite element algorithms for the computational simulation of type ii superconductors the algorithms are based on discretizations of a periodic ginzburg landau model periodicity is defined with respect to a non orthogonal lattice that is not necessarily aligned with the coordinate axes also the primary dependent variables employed in the model satisfy non standard quasi periodic boundary conditions after introducing the model we define finite element schemes derive error estimates of optimal order and present the results of some numerical calculations for a similar quality of simulation the resulting algorithms seem to be significantly less costly than are previously used numerical approximation methods

a linearized crank nicolson galerkin method for the ginzburg landau model the time dependent ginzburg landau model is used extensively in studying the nonequilibrium state of superconductivity the computer simulation of this model requires high performance computing power and reliable and efficient numerical methods to solve the ginzburg landau equtions in this paper a linearized crank nicolson galerkin method is proposed for solving these nonlinear and coupled partial differential equations the method uses the galerkin finite element approximation in spatial discretization and the crank nicolson implicit scheme in time discretization together with certain techniques which linearize and decouple the ginzburg landau equations while retaining the stabillity and accuracy of the crank nicolson scheme the proposed approach results in symmetric and positive definite matrix problems thus substantially improving the computational efficiency furthermore and even more important the proposed approach is suitable for large scale parallel computation numerical results from simulating the vortex dynamics of superconductivity by using the linearized crank nicolson galerkin method are presented

a new method for studying the vibration of non homogeneous membranes abstract we present a method to solve the helmholtz equation for a non homogeneous membrane with dirichlet boundary conditions at the border of arbitrary two dimensional domains the method uses a collocation approach based on a set of localized functions called little sinc functions which are used to discretize two dimensional regions we have performed extensive numerical tests and we have compared the results obtained with the present method with the ones available from the literature our results show that the present method is very accurate and that its implementation for general problems is straightforward

minimization of the ginzburg landau energy functional by a sobolev gradient trust region method we describe a generalized levenberg marquardt method for computing critical points of the ginzburg landau energy functional which models superconductivity the algorithm is a blend of a newton iteration with a sobolev gradient descent method and is equivalent to a trust region method in which the trust region radius is defined by a sobolev metric numerical test results demonstrate the method to be remarkably effective

radial solutions for the ginzburg landau equation with applied magnetic field abstract the purpose of this work is a systematic study of symmetric vortices for the ginzburg landau model of superconductivity along a cylinder with applied magnetic field parallel to its axis the ginzburg landau constant of the material and the degree d of the vortex are fixed for any given parameters r the radius of the cylinder and h the intensity of the applied magnetic field one can find a symmetric vortex a which satisfies the boundary conditions b r e i d and curl a b r he 3 it is then shown that symmetric vortices form a family depending continuously on two real parameters and c which describe the behaviour at the center of the vortex as the boundary conditions depend smoothly on those parameters one can distinguish two main connected domains of vortices the first one defined by the boundary conditions b r e i d and ra r b r is a zone of stability where remains increasing the second one defined by the boundary conditions b r e i d and ra r b r d is a zone where some instability can occur attention is focused on the smooth branch of vortices which separates those two domains it is indexed by the parameter running from some l 0 to and the limit as decreases to l corresponds to a berger and chen type vortex

analysis and approximation of optimal control problems for a simplified ginzburg landau model of superconductivity this paper is concerned with optimal control problems for a ginzburg landau model of superconductivity that is valid for high values of the ginzburg landau parameter and high external fields the control is of neumann type we first show that optimal solutions exist we then show that lagrange multipliers may be used to enforce the constraints and derive an optimality system from which optimal states and controls may be deduced then we define finite element approximations of solutions for the optimality system and derive error estimates for the approximations finally we report on some numerical results

schrodinger operators with non degenerately vanishing magnetic fields in bounded domains we establish an asymptotic estimate of the lowest eigenvalue bf of the schrodinger operator 2 bf with a magnetic field in a bounded 2 dimensional domain where curl f vanishes non degenerately and b is a large parameter our study is based on an analysis on an eigenvalue variation problem for the sturm liouville problem using the estimate we determine the value of the upper critical field for superconductors subject to non homogeneous applied magnetic fields and localize the nucleation of superconductivity

attractors and inertial manifolds for finite difference approximations of the complex ginzburg landau equation a semidiscrete spatial finite difference approximation and two fully discrete finite difference approximations to the complex ginzburg landau equation are considered in this paper the existence of an inertial manifold is proved inside a discrete h1 absorbing ball for the semi discrete approximation by showing that certain spectral properties hold for the linear operator and certain lipschitz properties hold for the nonlinear term convergence of the true and discrete inertial manifolds is then shown by considering the contraction mappings used to construct the inertial manifolds a fully implicit scheme is shown to form a continuous semigroup in a discrete l2 space and absorbing balls of radii independent of the spatial and temporal mesh sizes in discrete l2 and h1 spaces are constructed existence of a global attractor follows furthermore the existence of an inertial manifold is proved by showing that certain spectral and lipschitz properties hold for a semi implicit scheme existence of absorbing balls in the discrete space l2 of radii independent of the spatial and temporal mesh is proved existence of a global attractor follows

robust a priori and a posteriori error analysis for the approximation of allen cahn and ginzburg landau equations past topological changes a priori and a posteriori error estimates are derived for the numerical approximation of scalar and complex valued phase field models particular attention is devoted to the dependence of the estimates on a small parameter and to the validity of the estimates in the presence of topological changes in the solution that represents singular points in the evolution for typical singularities the estimates depend on the inverse of the parameter in a polynomial as opposed to exponential dependence of estimates resulting from a straightforward error analysis the estimates naturally lead to adaptive mesh refinement and coarsening algorithms numerical experiments illustrate the reliability and efficiency of this approach for the evolution of interfaces and vortices that undergo topological changes

a finite element method with mesh adaptivity for computing vortex states in fast rotating bose einstein condensates numerical computations of stationary states of fast rotating bose einstein condensates require high spatial resolution due to the presence of a large number of quantized vortices in this paper we propose a low order finite element method with mesh adaptivity by metric control as an alternative approach to the commonly used high order finite difference or spectral approximation methods the mesh adaptivity is used with two different numerical algorithms to compute stationary vortex states an imaginary time propagation method and a sobolev gradient descent method we first address the basic issue of the choice of the variable used to compute new metrics for the mesh adaptivity and show that refinement using simultaneously the real and imaginary part of the solution is successful mesh refinement using only the modulus of the solution as adaptivity variable fails for complicated test cases then we suggest an optimized algorithm for adapting the mesh during the evolution of the solution towards the equilibrium state considerable computational time saving is obtained compared to uniform mesh computations the new method is applied to compute difficult cases relevant for physical experiments large nonlinear interaction constant and high rotation rates

existence of surface smectic states of liquid crystals abstract the landau de gennes model of liquid crystals is a functional acting on wave functions order parameters and vector fields director fields in a specific asymptotic limit of the physical parameters we construct critical points such that the wave function order parameter is localized near the boundary of the domain and we determine a sharp localization of the boundary region where the wave function concentrates furthermore we compute the asymptotics of the energy of such critical points along with a boundary energy that may serve in localizing the director field in physical terms our results prove the existence of a surface smectic state

superconductivity near the normal state under the action of electric currents and induced magnetic fields in mathbb r 2 we consider the linearization of the time dependent ginzburg landau system near the normal state we assume that an electric current is applied through the sample which captures the whole plane inducing thereby a magnetic field we show that independently of the current the normal state is always stable using fourier analysis the detailed behaviour of solutions is obtained as well relying on semi group theory we then obtain the spectral properties of the steady state elliptic operator

a fast semi implicit finite difference method for the tdgl equations abstract we propose a finite difference algorithm for solving the time dependent ginzburg landau tdgl equation coupled to the appropriate maxwell equation the time derivatives are discretized using a second order semi implicit scheme which for intermediate values of the ginzburg landau parameter allows time steps two orders of magnitude larger than commonly used in explicit schemes we demonstrate the use of the method by solving a fully three dimensional problem of a current carrying wire with longitudinal and transverse magnetic fields

existence of periodic solutions for ginzburg landau equations of superconductivity abstract in an earlier paper b wang 1999 j math anal appl 232 394 412 the author proved the existence of periodic solutions to the ginzburg landau equations of superconductivity in two dimensions here we will show that the ginzburg landau equations admit periodic solutions in three dimensions as well we also prove the existence of periodic solutions to the phase lock equations introduced by the author in press j nonlinear anal

on the solutions of the one dimensional ginzburg landau equations for superconductivity this paper gives a complete description of the solutions of the one dimensional ginzburg landau equations which model superconductivity phenomena in infinite slabs we investigate this problem over the entire range of physically important parameters a the size of the slab the ginzburg landau parameter and h0 the exterior magnetic field we do extensive numerical computations using the software auto and determine the number symmetry and stability of solutions for all values of the parameters in particular our experiments reveal the existence of two key points in parameter space which play a central role in the formation of the complicated patterns by means of bifurcation phenomena our global description also allows us to separate the various physically important regimes to classify previous results in each regime according to the values of the parameters and to derive new open problems in addition our investigation provides new insight into the problem of differentiating between the types of superconductors in terms of the parameters

boundary concentration for eigenvalue problems related to the onset of superconductivity we examine the asymptotic behavior of the eigenvalue h and corresponding eigenfunction associated with the variational problem in the regime h 1 here a is any vector field with curl equal to 1 the problem arises within the ginzburg landau model for superconductivity with the function h yielding the relationship between the critical temperature vs applied magnetic field strength in the transition from normal to superconducting state in a thin mesoscopic sample with cross section omega subset r 2 we first carry out a rigorous analysis of the associated problem on a half plane and then rigorously justify some of the formal arguments of bs obtaining an expansion for while also proving that the first eigenfunction decays to zero somewhere along the sample boundary when is not a disc for interior decay we demonstrate that the rate is exponential

plane wave diffraction by an impedance step plane wave diffraction by two half planes joined by a step is studied in the case where the half planes and the step are characterized by different surface impedances the diffraction problem is first reduced to a modified wiener hopf equation of the second kind whose solution contains an infinite set of constants satisfying an infinite system of linear algebraic equations a numerical solution of this system is obtained for various values of the surface impedances and the height of the step through which the effect of these parameters on the diffraction phenomenon is studied

directional curl spaces and applications to the meissner states of anisotropic superconductors the directional curl spaces are introduced which consist of vector fields that have directional curl along the given unit length vector fields the basic properties of these spaces are derived including the trace theorem and the representation of the dual spaces the directional curl spaces are used to analyze the asymptotic behavior of the meissner states of strongly anisotropic superconductors where the limiting energy functionals are non coercive in the usual sobolev spaces

magnetic line source diffraction by an impedance step diffraction of a magnetic line source by an impedance step joined by two half planes is studied in the case where the half planes and step are characterized by different surface impedances the problem is solved using wiener hopf technique and fourier transform the scattered field in the far zone is determined by the method of steepest descent graphical results for the solution has also been presented it is observed that if the source is shifted to a large distance these results differ from those of by a multiplicative factor to the part of the scattered field containing the effects of incident and reflected waves

analysis and convergence of a covolume approximation of the ginzburg landau model of superconductivity in this paper we present the mathematical analysis of a covolume method for the approximations of the ginzburg landau gl model for superconductivity a nice feature of this approach is that the gauge invariance properties are retained in discrete approximations based on triangular grids we also use properties of discrete vector fields to study issues such as the gauge choices and their enforcement

vibration of non homogeneous rectangular membranes with arbitrary interfaces abstract the study deals with the generalized solution of the title problem the free vibration problem of a rectangular membrane with partial domains each of uniform density and arbitrary interface is tackled previous studies by other researchers include straight parallel to the borders and oblique interfaces and bent ones the solution is found by means of a direct variational method with a series composed with a complete set of functions two alternative sets are explored trigonometric and power series such series are uniformly convergent to the exact solution the approach is straightforward and very efficient from the computational viewpoint a determinant factorization method is employed to automatically eliminate eventual spurious frequencies the well known analogy between plates and membranes does not hold in this problem and a demonstration is included diverse illustrations are worked out as the cases of an oblique straight line interface and an open curve line which divides the membrane in two domains each of different density are first presented also a rectangular membrane with an interior closed domain is stated and the numerical example of a circular interior zone is included comparison between the two alternatives and with other authors results show excellent agreement in all cases the computational cost is very low