asymptotic limits of the full navier stokes fourier poisson system abstract the combined quasineutral and zero electron mass limit of the full navier stokes fourier poisson system in the torus t d d 1 is considered it is showed that for well prepared initial data the weak solutions of the full navier stokes fourier poisson system converge to the strong solution of the incompressible navier stokes equations as long as the latter exists moreover the convergence rates are obtained

some solutions of the linear and nonlinear klein gordon equations using the optimal homotopy asymptotic method abstract we investigate the effectiveness of the optimal homotopy asymptotic method oham in solving time dependent partial differential equations to this effect we consider the homogeneous non homogeneous linear and nonlinear klein gordon equations with boundary conditions the results reveal that the method is explicit effective and easy to use

reciprocity laws and galois representations recent breakthroughs suppose f x is a monic irreducible polynomial with integer coefficients if p is a prime number then reducing the coefficients of f x modulo p gives a new polynomial fp x which may be reducible we say that f x is split modulo p if fp x is the product of distinct linear factors example 1 1 1 the polynomial f x x2 1 is split modulo 5 since f5 x x 2 x 3 mod 5 but it is not split modulo 7 since f7 x is irreducible nor is it split modulo 2 since f2 x x 1 2 mod 2 has a repeated factor the first few p for which x2 1 is split modulo p are 5 13 17 29 37 41 53

on singular limits of mean field equations mean field equations arise as steady state versions of convection diffusion systems where the convective field is determined by solution of a poisson equation whose right hand side is affine in the solutions of the convection diffusion equations in this paper we consider the repulsive coupling case for a system of two convection diffusion equations for general diffusivities we prove the existence of a unique solution of the mean field equation by a variational analysis of a saddle point problem usually without coercivity also we analyze the small debye length limit and prove convergence to either the so called charge neutral case or to a double obstacle problem for the limiting potential depending on the data

quasi neutral limit for a model of viscous plasma we perform a rigorous analysis of the quasi neutral limit for a model of viscous plasma represented by the navier stokes poisson system of equations it is shown that the limit problem is the navier stokes system describing a barotropic fluid flow with the pressure augmented by a component related to the nonlinearity in the original poisson equation

vanishing cross diffusion limit in a keller segel system with additional cross diffusion abstract keller segel systems in two and three space dimensions with an additional cross diffusion term in the equation for the chemical concentration are analyzed the cross diffusion term has a stabilizing effect and leads to the global in time existence of weak solutions the limit of vanishing cross diffusion parameter is proved rigorously in the parabolic elliptic and parabolic parabolic cases when the signal production is sublinear the existence of global in time weak solutions as well as the convergence of the solutions to those of the classical parabolic elliptic keller segel equations are proved the proof is based on a reformulation of the equations eliminating the additional cross diffusion term but making the equation for the cell density quasilinear for superlinear signal production terms convergence rates in the cross diffusion parameter are proved for local in time smooth solutions since finite time blow up is possible the proof is based on careful h s estimates and a variant of the gronwall lemma numerical experiments in two space dimensions illustrate the theoretical results and quantify the shape of the cell aggregation bumps as a function of the cross diffusion parameter

quasineutral limit of a time dependent drift diffusion poisson model for p n junction semiconductor devices abstract in this paper the vanishing debye length limit of the bipolar time dependent drift diffusion poisson equations modelling insulated semiconductor devices with p n junctions i e with a fixed bipolar background charge is studied for sign changing and smooth doping profile with good boundary conditions the quasineutral limit zero debye length limit is performed rigorously by using the multiple scaling asymptotic expansions of a singular perturbation analysis and the carefully performed classical energy methods the key point in the proof is to introduce a density transform and two weighted liapunov type functionals first and then to establish the entropy production integration inequality which yields the uniform estimate with respect to the scaled debye length the basic point of the idea involved here is to control strong nonlinear oscillation by the interaction between the entropy and the entropy dissipation

symmetric and arbitrarily high order birkhoff hermite time integrators and their long time behaviour for solving nonlinear klein gordon equations abstract the klein gordon equation with nonlinear potential occurs in a wide range of application areas in science and engineering its computation represents a major challenge the main theme of this paper is the construction of symmetric and arbitrarily high order time integrators for the nonlinear klein gordon equation by integrating birkhoff hermite interpolation polynomials to this end under the assumption of periodic boundary conditions we begin with the formulation of the nonlinear klein gordon equation as an abstract second order ordinary differential equation ode and its operator variation of constants formula we then derive a symmetric and arbitrarily high order birkhoff hermite time integration formula for the nonlinear abstract ode accordingly the stability convergence and long time behaviour are rigorously analysed once the spatial differential operator is approximated by an appropriate positive semi definite matrix subject to suitable temporal and spatial smoothness a remarkable characteristic of this new approach is that the requirement of temporal smoothness is reduced compared with the traditional numerical methods for pdes in the literature numerical results demonstrate the advantage and efficiency of our time integrators in comparison with the existing numerical approaches

a family of calabi yau varieties and potential automorphy we prove potential modularity theorems for l adic representations of any dimension from these results we deduce the sato tate conjecture for all elliptic curves with nonintegral j invariant defined over a totally real field

a decomposition method for solving the nonlinear klein gordon equation in this paper adomian s decomposition scheme is presented as an alternate method for solving the nonlinear klein gordon equation the method is demonstrated by several examples comparing the scheme with existing collocation finite difference and finite element techniques shows that the present approach is highly accurate and converges rapidly

an algorithm for solving boundary value problems in this paper a numerical algorithm based on the decomposition technique is presented for solving a class of nonlinear boundary value problems the method is implemented for well known examples including troesch s and bratu s problems which have been extensively studied the scheme is shown to be highly accurate and only a few terms are required to obtain accurate computable solutions

analysis and comparison of numerical methods for the klein gordon equation in the nonrelativistic limit regime we analyze rigourously error estimates and compare numerically temporal spatial resolution of various numerical methods for solving the klein gordon kg equation in the nonrelativistic limit regime involving a small parameter 0 varepsilon ll 1 which is inversely proportional to the speed of light in this regime the solution is highly oscillating in time i e there are propagating waves with wavelength of o varepsilon 2 and o 1 in time and space respectively we begin with four frequently used finite difference time domain fdtd methods and obtain their rigorous error estimates in the nonrelativistic limit regime by paying particularly attention to how error bounds depend explicitly on mesh size h and time step as well as the small parameter varepsilon based on the error bounds in order to compute correct solutions when 0 varepsilon ll1 the four fdtd methods share the same varepsilon scalability tau o varepsilon 3 then we propose new numerical methods by using either fourier pseudospectral or finite difference approximation for spatial derivatives combined with the gautschi type exponential integrator for temporal derivatives to discretize the kg equation the new methods are unconditionally stable and their varepsilon scalability is improved to o 1 and tau o varepsilon 2 for linear and nonlinear kg equations respectively when 0 varepsilon ll1 numerical results are reported to support our error estimates

quasi neutral limit of the full navier stokes fourier poisson system abstract the quasi neutral limit of the full navier stokes fourier poisson system in the torus t d d 1 is considered we rigorously prove that as the scaled debye length goes to zero the global in time weak solutions of the full navier stokes fourier poisson system converge to the strong solution of the incompressible navier stokes equations as long as the latter exists in particular the effect of large temperature variations is taken into account

serre s modularity conjecture ii we provide proofs of theorems 4 1 and 5 1 of khare and wintenberger invent math doi 10 1007 s00222 009 0205 7 2009

exact solutions for the generalized klein gordon equation via a transformation and exp function method and comparison with adomian s method in this paper a suitable transformation and a so called exp function method are used to obtain different types of exact solutions for the generalized klein gordon equation these exact solutions are in full agreement with the previous results obtained in refs sirendaoreji auxiliary equation method and new solutions of klein gordon equations chaos solitons fractals 31 4 2007 943 950 huiqun zhang extended jacobi elliptic function expansion method and its applications communications in nonlinear science and numerical simulation 12 5 2007 627 635 one of these exact solutions is compared with the approximate solutions obtained by the modified decomposition method accurate numerical results for a wider range of time are obtained after using different types of adm pade approximation our results show that the exp function method is very effective in finding exact solutions for the problem considered while the modified decomposition method is very powerful in finding numerical solutions with good accuracy for nonlinear pde without any need for a transformation or perturbation

quasi neutral limit to the drift diffusion models for semiconductors with physical contact insulating boundary conditions abstract in this paper the limit of vanishing debye length in a bipolar drift diffusion model for semiconductors with physical contact insulating boundary conditions is studied in one dimensional case the quasi neutral limit zero debye length limit is proved by using the asymptotic expansion methods of singular perturbation theory and the classical energy methods our results imply that one kind of the new and interesting phenomena in semiconductor physics occurs

the limiting problem of the drift diffusion poisson model with discontinuous p n junctions abstract in this paper the authors consider the limiting problem of the drift diffusion poisson model for semiconductors different from previous papers the model considered involve some special doping profiles d which have the property that the function is allowed to have a jump discontinuity and sign changing property but d 2 is required to be lipschitz continuous the existence uniqueness and large time asymptotic behavior of the global in time solutions are given

quasi neutral limit of a nonlinear drift diffusion model for semiconductors the limit of the vanishing debye length the charge neutral limit in a nonlinear bipolar drift diffusion model for semiconductors without a pn junction i e with a unipolar background charge is studied the quasi neutral limit zero debye length limit is determined rigorously by using the so called entropy functional which yields appropriate uniform estimates

the iwasawa main conjectures for gl2 we prove the one two and three variable iwasawa greenberg main conjectures for a large class of modular forms that are ordinary with respect to an odd prime p the method of proof involves an analysis of an eisenstein ideal for ordinary hida families for gu 2 2

renormalization of cellular automata and self similarity we study self similarity in one dimensional probabilistic cellular automata pca using the renormalization technique we introduce a general framework for algebraic construction of renormalization groups rg on cellular automata and apply it to exhaustively search the rule space for automata displaying dynamic criticality previous studies have shown that there exists several exactly renormalizable deterministic automata we show that the rg fixed points for such self similar ca are unstable in all directions under renormalization this implies that the large scale structure of self similar deterministic elementary cellular automata is destroyed by any finite error probability as a second result we show that the only non trivial critical pca are the different versions of the well studied phenomenon of directed percolation we discuss how the second result supports a conjecture regarding the universality class for dynamic criticality defined by directed percolation

vacuum solution and quasineutral limit of semiconductor drift diffusion equation article history in recent years the theory of semiconductor device modeling has become an area of increasing in terest in applied mathematics semiconductor drift diffusion equation is most widely used to describe the flow of electrons in the conduction band and of holes in the valence band of semiconductor mate rial influenced by electric field the model consists of the continuity equations for electron and hole densities n p respectively and of the poisson equation for the electric potential v

a singularly perturbed boundary value problem modelling a semiconductor device this paper is concerned with the static one dimensional modelling of a semiconductor device namely the pn junction when a bias is applied the governing equations are the well known equations describing carrier transport in a semiconductor which consist of a system of ordinary differential equations subject to boundary conditions imposed at the contacts because of the different orders of magnitude of the solution components at the boundaries we scale the components individually and obtain a singular perturbation problem we analyse the equilibrium case zero bias applied and set up approximate models posed as singularly perturbed second order equations by neglecting the hole and electron current densities this makes sense for small forward bias and for moderate reverse bias for the full problem we prove an a priori estimate on the number of electron hole carrier pairs and derive asymptotic expansions as the perturbation parameter tends to zero by setting up the reduced system and the boundary

convergence of the navier stokes poisson system to the incompressible navier stokes equations the quasineutral limit of the navier stokes poisson system in the whole space rd d 1 and in the torus td is studied in this paper it is shown that for the well prepared initial data the global weak solution of the navier stokes poisson system converges strongly to the strong solution of the incompressible navier stokes equations

an asymptotic anaylsis of a transient p n junction model in this paper we carry out an asymptotic analysis of the system of differential equations describing the transient behavior of a p n junction device i e a diode we determine the different time scales present in the equations and investigate which of them actually occur in physical situations we derive asymptotic expansions of the solution and perform some numerical experiments

a uniformly accurate ua multiscale time integrator pseudospectral method for the dirac equation in the nonrelativistic limit regime we propose and rigourously analyze a multiscale time integrator fourier pseudospectral mti fp method for the dirac equation with a dimensionless parameter varepsilon in 0 1 which is inversely proportional to the speed of light in the nonrelativistic limit regime i e 0 varepsilon ll 1 the solution exhibits highly oscillatory propagating waves with wavelength o varepsilon 2 and o 1 in time and space respectively due to the rapid temporal oscillation it is quite challenging in designing and analyzing numerical methods with uniform error bounds in varepsilon in 0 1 we present the mti fp method based on properly adopting a multiscale decomposition of the solution of the dirac equation and applying the exponential wave integrator with appropriate numerical quadratures by a careful study of the error propagation and using the energy method we establish two independent error estimates via two different mathematical approaches as h m 0 frac tau 2 varepsilon 2 and h m 0 tau 2 varepsilon 2 where h is the mesh size tau is the time step and m 0 depends on the regularity of the solution these two error bounds immediately imply that the mti fp method converges uniformly and optimally in space with exponential convergence rate if the solution is smooth and uniformly in time with linear convergence rate at o tau for all varepsilon in 0 1 and optimally with quadratic convergence rate at o tau 2 in the regimes when either varepsilon o 1 or 0 varepsilon lesssim tau numerical results are reported to demonstrate that our error estimates are optimal and sharp finally the mti fp method is applied to study numerically the convergence rates of the solution of the dirac equation to those of its limiting models when varepsilon to0

quasi neutral limit and the boundary layer problem of planck nernst poisson navier stokes equations for electro hydrodynamics abstract quasi neutral limit and the boundary layer problem of the two three dimensional incompressible planck nernst poisson navier stokes equations for electro hydrodynamics with the general mobilities of two kinds of charges and in the bounded domain is studied for the generally smooth doping profile and the physical case that the mobilities of two kinds of charges is different quasi neutral limit with a subtle boundary layer structure is rigorously proved by constructing one new weighted entropy functional coupled with multi scaling approximating expansion techniques

a spline collocation approach for the numerical solution of a generalized nonlinear klein gordon equation in this paper a finite element collocation approach using cubic b splines is employed for the numerical solution of a generalized form of the nonlinear klein gordon equation the efficiency of the method is tested on a number of examples that represent special cases of the extended equation including the sine gordon equation the numerical results are compared with existing numerical and analytic solutions and the outcomes confirm that the scheme yields accurate and reliable results even when few nodes are used at the time levels

fourier collocation method for solving nonlinear klein gordon equation a fourier collocation scheme is proposed for solving the periodic problem of nonlinear klein gordon equation it keeps the energy discretely and is shown to be stable or generalized stable its convergence is investigated numerical results are also presented

most odd degree hyperelliptic curves have only one rational point consider the smooth projective models c of curves y 2 f x with f x 2 z x monic and separable of degree 2g 1 we prove that for g 3 a positive fraction of these have only one rational point the point at innity we prove a lower bound on this fraction that tends to 1 as g 1 finally we show that c q can be algorithmically computed for such a fraction of the curves the method can be summarized as follows using p adic analysis and an idea of mccallum we develop a reformulation of chabauty s method that shows that certain computable conditions imply c q 1 on the other hand using further p adic analysis the theory of arithmetic surfaces a new result on torsion points on hyperelliptic curves and crucially the bhargava gross theorems on the average number and equidistribution of nonzero 2 selmer group elements we prove that these conditions are often satised for p 2

solving integral equations by reconstruction in isomorphic taylor coefficient spaces a new method for numerically solving integral equations is presented several examples are shown to demonstrate the accuracy of the procedure using the procedure one can in principle solve any properly constructed integral equation for solutions in the analytic domain

an exponential wave integrator sine pseudospectral method for the klein gordon zakharov system an exponential wave integrator sine pseudospectral method is presented and analyzed for discretizing the klein gordon zakharov kgz system with two dimensionless parameters 0 varepsilon le 1 and 0 gamma le 1 which are inversely proportional to the plasma frequency and the speed of sound respectively the main idea in the numerical method is to apply the sine pseudospectral discretization for spatial derivatives followed by using an exponential wave integrator for temporal derivatives in phase space the method is explicit symmetric in time and it is of spectral accuracy in space and second order accuracy in time for any fixed varepsilon varepsilon 0 and gamma gamma 0 in the mathcal o 1 plasma frequency and speed of sound regime i e varepsilon mathcal o 1 and gamma mathcal o 1 we establish rigorously error estimates for the numerical method in the energy space h 1 times l 2 we also study numerically the resolution of the method in the simultaneous high plasma fr

a new approach to the cubic schrodinger equation an application of the decomposition technique abstract a decomposition method is implemented for solving the cubic schrodinger equation

solving the klein gordon equation by means of the homotopy analysis method an analytic technique namely the homotopy analysis method is applied to solve the nonlinear travelling waves governed by the klein gordon equation the phase speed and the solution which are dependent on the amplitude a are given and valid in the whole region 0

how many rational points does a random curve have a large part of modern arithmetic geometry is dedicated to or motivated by the study of rational points on varieties for an elliptic curve over q the set of rational points forms a finitely generated abelian group the ranks of these groups when ranging over all elliptic curves are conjectured to be evenly distributed between rank 0 and rank 1 with higher ranks being negligible we will describe these conjectures and discuss some results on bounds for average rank highlighting the recent work of bhargava and shankar

stability and convergence of trigonometric integrator pseudospectral discretization for n coupled nonlinear klein gordon equations abstract this work serves as an improvement on a recent paper dong 2013 9 in which the n coupled nonlinear klein gordon equations were solved numerically by a fully explicit trigonometric integrator fourier pseudospectral tifp method this tifp method is second order accurate in time and spectral order accurate in space however in the previous work there was an absence of rigorous stability and convergence analysis moreover numerical studies in this work suggest that this tifp method suffers from a stability condition o h and h refer to time step and space mesh size to relax such a restriction while keeping the convergence properties and explicitness we propose two modified tifp methods motivated by the mollified impulse and gautschi type integrators for oscillatory odes for the modifications considered here linear stability and rigorous error estimates in the energy space are carried out which are the main achievements gained in this work meanwhile numerical results are also presented ideas of this work also suggest a general framework for proposing and analyzing efficient numerical methods for coupled wave type equations

on the solution of coupled h like equations of chandrasekhar in this paper a decomposition algorithm is presented for the solution of the coupled integral h like equations of chandrasekhar that arise in transport theory an important feature of this numerical technique is its rapid convergence and the high degree of accuracy by which it approximates a solution using only few terms of its iterative scheme

an algorithm for solving bond pricing problem the nonlinear bond pricing problem has been extensively studied in the literature since an analytical solution is not readily available we will seek to find an approximate solution we will present a decomposition method due to adomian and show how to obtain a reasonable numerical solution to the bond pricing problem

initial time layer problem for quantum drift diffusion model we study the initial time layer problem to the quantum drift diffusion model the limit of vanishing debye length is established by entropy method and compactness argument as a byproduct the global weak solution is also obtained for this model

the sine gordon equation in the finite line a rather complete numerical study of the perturbed and nonperturbed sine gordon equation in one dimensional cartesian coordinates and finite domains is presented the numerical study is based on five three point linearly implicit finite difference methods of different spatial accuracy for the unperturbed sge the accuracy of these five methods is found to be very sensitive to the implicitness parameter less sensitive to the spatial order of accuracy and almost no sensitive to the time step for second order accurate temporal discretizations the largest errors of the five methods were found to occur at the front of the soliton solution of the unperturbed sge but the techniques were very accurate for long time computations of the unperturbed sge in both infinite domains and in finite domains upon many collisions of the kinks and antikinks with the boundaries despite the fact that they do not preserve a discrete energy it is shown that the largest values of the kinetic and total energies and the smallest values of the strain and potential energies occur when solitons and antisolitons strike on the boundaries subject to homogeneous neumann conditions and the four energies remain constant between collisions with the boundaries for the malomed s perturbation one of the fronts of the soliton soliton doublet and soliton antisoliton doublet is trapped at one boundary and the other one undergoes a change in amplitude and its front describes a curved trajectory the soliton antisoliton breather oscillates in a damped manner and slowly drifts towards one boundary the soliton antisoliton soliton soliton doublet and soliton antisoliton doublet of the unperturbed sge are robust under kivshar and malomed s perturbations whereas the soliton antisoliton breather loses its integrity becomes a kink antikink pair and then yields a chaotic behaviour in both space and time small amplifications proportional to the first order temporal derivative of the amplitude do not alter substantially the initial dynamics of the soliton antisoliton soliton soliton doublet and soliton antisoliton doublet but their amplitudes exhibit oscillatory behaviour of increasing amplitude at large times however small amplifications cause first a transition from the soliton antisoliton breather to the kink antikink solution a linearized stability analysis of the spatially homogeneous solutions of the unperturbed sge is performed and analytical solutions are obtained by means of the method of separation of variables for both homogeneous dirichlet and neumann boundary conditions and the results show that the solution corresponding to a zero amplitude is oscillatory in both space and time whereas that corresponding to an amplitude equal to p may exhibit exponential or linear growth depending on the length of the domain if either of these growths is present it is shown that the evolution of small initial perturbations may result in homoclinic crossings and chaotic behaviour in both space and time in the absence of perturbations in the sge and the existence of an attractor when perturbations of the malomed s type are present in either case the solution may exhibit certain spatial symmetries or antisymmetries at large times finally the results presented in this paper indicate that the accurate computation of homoclinic orbits of the sge is not only controlled by the spatial accuracy of the numerical method but also by its energy preserving characteristics and perhaps by the number of grid points used in the numerical technique

a new approach to bratu s problem in this paper a laplace transform decomposition numerical algorithm is introduced for solving bratu s problem the numerical scheme is based on the application of laplace transform integral operator to the differential equation the nonlinear term is then decomposed and an iterative algorithm is constructed for the determination of the infinite series solution the technique is illustrated with two numerical examples and the results show that the method converges rapidly and approximates the exact solution very accurately using only few iterates of the recursive scheme

a singular perturbation analysis of the transient semiconductor device equations this paper is concerned with the asymptotic analysis of the transient semiconductor device equations by appropriately scaling the equations the semiconductor device problem is reformulated as a si

quasi neutral limit to the drift diffusion models for semiconductors with physical contact insulating boundary conditions and the general sign changing doping profile abstract the quasi neutral limit in a bipolar drift diffusion model for semiconductors with physical contact insulating boundary conditions the general sign changing doping profile and general initial data which allow the presence of the left and right boundary layers and the initial layers is studied in the one dimensional case the dynamic structure stability of the solution with respect to the scaled debye length is proven by the asymptotic analysis of singular perturbation and the entropy energy method the key point of the proof is to use sufficiently the fact that the length of the boundary layer is very small in a short time period

potential automorphy and change of weight we prove an automorphy lifting theorem for l adic representations where we impose a new condition at l which we call potentential diagonalizability this result allows for change of weight and seems to be substantially more exible than previous theorems along the same lines we derive several applications for instance we show that any irreducible totally odd essentially self dual regular weakly compatible system of l adic representations of the absolute galois group of a totally real eld is potentially automorphic and hence is pure and its l function has meromorphic continuation to the whole complex plane and satises the expected functional equation

error estimates in the energy space for a gautschi type integrator spectral discretization for the coupled nonlinear klein gordon equations we propose and analyze an efficient and accurate numerical method for solving the coupled nonlinear klein gordon equations the method is based on the application of a gautschi type exponential integrator in time combined with sine spectral discretization in space the main results achieved in this paper are the rigorous error estimates in the energy space h 1 h 1 for the proposed scheme numerical tests are reported and agree with the error estimates quite well

evaluation of mixed crank nicolson scheme and tau method for the solution of klein gordon equation abstract numerical method based on the crank nicolson scheme and the tau method is proposed for solving nonlinear klein gordon equation nonlinear klein gordon equation is reduced by crank nicolson scheme to the system of ordinary differential equations then tau method is used to solve this system by using interpolating scaling functions and operational matrix of derivative the order of convergence is proposed and some numerical examples are included to demonstrate the validity and applicability of the technique the method is easy to implement and produces accurate results

an efficient high order explicit scheme for solving hamiltonian nonlinear wave equations in this work we propose and analyze a novel high order explicit scheme for efficiently solving hamiltonian nonlinear wave equations the new explicit scheme is based on the blend of a fourth order finite difference scheme for spatial discretization and a multidimensional extended runge kutta nystrom erkn method for time integration respectively the conservation law of the semi discrete energy is established the stability and convergence of the semidiscretization are examined the results of numerical experiments show that the blend of the finite difference approximation and multidimensional erkn method gives an efficient high order explicit scheme for hamiltonian nonlinear wave equations in comparison with some existing methods

analysis of a finite element method for the drift diffusion semiconductor device equations the multidimensional case an explicit finite element method for numerically solving the drift diffusion semiconductor device equations in two space dimensions is analyzed the method is based on the use of a mixed finite element method for the approximation of the electric field and a discontinuous upwinding finite element method for the approximation of the electron and hole concentrations the mixed method gives an approximate electric field in the precise form needed by the discontinuous method which is trivially conservative and fully parallelizable it is proven that the method produces uniformly bounded concentrations and electric fields and that it converges to the exact solution provided there is a convergent subsequence of the electron concentrations numerical simulations are presented that display the performance of the method and indicate the behavior of the solution

convergence of a finite element method for the drift diffusion semiconductor device equations the zero diffusion case in this paper a new explicit finite element method for numerically solving the drift diffusion semiconductor device equations is introduced and analyzed the method uses a mixed finite element method for the approximation of the electric field a finite element method using discontinuous finite elements is used to approximate the concentrations which may display strong gradients the use of discontinuous finite elements renders the scheme for the concentrations trivially conservative and fully parallelizable in this paper we carry out the analysis of the model method which employs a continuous piecewise linear approximation to the electric field and a piecewise constant approximation to the electron concentration in a model problem namely the so called unipolar case with the diffusion terms neglected the resulting system of equations is equivalent to a conservation law whose flux the electric field depends globally on the solution the concentration of electrons by exploiting the similarities of this system with classical scalar conservation laws the techniques to analyze the monotone schemes for conservation laws are adapted to the analysis of the new scheme the scheme considered as a scheme for the electron concentration is shown to satisfy a maximum principle and to be total variation bounded its convergence to the unique weak solution is proven numerical experiments displaying the performance of the scheme are shown

arbitrarily high order time stepping schemes based on the operator spectrum theory for high dimensional nonlinear klein gordon equations abstract in this paper we explore arbitrarily high order lagrange collocation type time stepping schemes for effectively solving high dimensional nonlinear klein gordon equations with different boundary conditions we begin with one dimensional periodic boundary problems and first formulate an abstract ordinary differential equation ode on a suitable infinity dimensional function space based on the operator spectrum theory we then introduce an operator variation of constants formula which is essential for the derivation of our arbitrarily high order lagrange collocation type time stepping schemes for the nonlinear abstract ode the nonlinear stability and convergence are rigorously analysed once the spatial differential operator is approximated by an appropriate positive semi definite matrix under some suitable smoothness assumptions with regard to the two dimensional dirichlet or neumann boundary problems our new time stepping schemes coupled with discrete fast sine cosine transformation can be applied to simulate the two dimensional nonlinear klein gordon equations effectively all essential features of the methodology are present in one dimensional and two dimensional cases although the schemes to be analysed lend themselves with equal to higher dimensional case the numerical simulation is implemented and the numerical results clearly demonstrate the advantage and effectiveness of our new schemes in comparison with the existing numerical methods for solving nonlinear klein gordon equations in the literature

nonconforming quadrilateral finite element method for a class of nonlinear sine gordon equations nonconforming quadrilateral finite element method fem of the two dimensional nonlinear sine gordon equation is studied for semi discrete and crank nicolson fully discrete schemes respectively firstly we prove a special feature of a new arbitrary quadrilateral element named modified quasi wilson element i e the consistency error is of order o h 2 h denotes the mesh size in h 1 norm which leads to optimal order error estimate and superclose result with order o h 2 for the semi discrete scheme through a different approach from the existing literature secondly because the consistency error estimate of the new modified quasi wilson element can reach a staggering o h 3 order two orders higher than that of interpolation error the optimal order error estimates of crank nicolson fully discrete scheme are obtained on arbitrary quadrilateral meshes with ritz projection moreover a superclose result in h 1 norm is presented on generalized rectangular meshes by a new technique thirdly the global superconvergence results of h 1 norm for both semi discrete and fully discrete schemes are derived on rectangular meshes with interpolated postprocessing technique finally a numerical test is carried out to verify the theoretical analysis