discrete time orthogonal spline collocation methods for schrodinger equations in two space variables two discrete time orthogonal spline collocation schemes are formulated and analyzed for solving the linear time dependent schrodinger equation in two space variables these are crank nicolson and alternating direction implicit adi schemes employing c1 piecewise polynomial spaces of arbitrary degree ge 3 in each space variable the stability of the schemes is examined and optimal order a priori l2 and h1 error estimates at each time step are derived parallel implementation of the adi scheme is discussed

a new finite variable difference method with application to locally exact numerical scheme a new finite variable difference method fvdm on finite differencing is presented the essence of this method consists in determining the optimum spatial difference such that the total variance of the solution is a minimum under the condition that characteristic roots of the resulting difference equation are always nonnegative to ensure the numerical stability the present fvdm is applied to the locally exact numerical scheme lens the optimum spatial difference of the lens is derived in terms of local mesh reynolds numbers by using this optimum spatial difference the numerical accuracy of the lens for the linear convection diffusion equations is increased without numerical oscillations for all mesh reynolds numbers the present study suggests that an optimum spatial difference from the viewpoint of numerical stability and accuracy exists according to the numerical schemes

galerkin alternating direction method for a kind of three dimensional nonlinear hyperbolic problems a kind of three dimensional nonlinear hyperbolic equation is firstly transformed into a first order system of equations then the galerkin alternating direction procedure for the system is derived the error estimates of the procedure in h 1 norm and l 2 norm respectively are obtained by using the theory and techniques of priori estimate of differential equations the numerical experiment is also given to support the theoretical analysis comparing the results of numerical example with the theoretical analysis they are coincided

a stable and accurate convective modelling procedure based on quadratic upstream interpolation a convective modelling procedure is presented which avoids the stability problems of central differencing while remaining free of the inaccuracies of numerical diffusion associated with upstream differencing for combined convection and diffusion the number of operations at each grid point is comparable to that of standard upstream pluscentral differencing however highly accurate solutions can be obtained with a grid spacing much larger than that required by conventional methods for comparable accuracy with obvious practical advantaged in terms of both speed and storage the algorithm is based on a conservative control volume formulation with cell wall values of each field variable written in terms of a quadratic interpolation using in any one coordinate direction the two adjacent nodal values together with the value at the next upstream node this results in a convective differencing scheme with greater formal accuracy than central differencing while retaining the basic stable convective sensitivity property of upstream weighted schemes the consistent treatment of diffusion terms is equivalent to central differencing with careful modelling numerical boundary conditions are not troublesome some idealized problems are studied showing the practical advantages of the method over other schemes in comparison with exact solutions an application to a complex unsteady two dimensional flow is briefly discussed

approximating common fixed points of nonexpansive mappings and nonspreading mappings in a hilbert space abstract in this paper we obtain some fundamental properties for nonspreading mappings in a hilbert space further we study the approximation of common fixed points of nonexpansive mappings and nonspreading mappings in a hilbert space

crank nicolson adi galerkin finite element method for two dimensional fractional fitzhugh nagumo monodomain model in this paper a two dimensional fractional fitzhugh nagumo monodomain model 2d ffhnmm with zero dirichlet boundary condition is considered the model consists of a coupled two dimensional space fractional nonlinear reaction diffusion model 2d sfnrdm and an ordinary differential equation the 2d sfnrdm and ordinary differential equation are decoupled at each time step a new crank nicolson alternating direction implicit adi galerkin finite element method for the 2d sfnrdm is developed the stability and convergence of the numerical method are discussed finally some numerical examples on 2d sfnrdm and 2d ffhnmm are given for verification of our theoretical analysis

an alternating direction implicit galerkin finite element method for the distributed order time fractional mobile immobile equation in two dimensions abstract in this paper we shall present the alternating direction implicit adi galerkin finite element method fem for solving the distributed order time fractional mobile immobile equation in two dimensions in the time direction the backward euler method is used to deal with the temporal first order derivative and the weighted and shifted grunwald formula is employed to discretize the distributed order time fractional derivative galerkin fem is used for discretization of the spatial direction and then an adi galerkin finite element scheme is constructed the stability and l 2 norm convergence are proved several numerical experiments are provided to verify our theoretical analysis

on uniform opial condition and uniform kadec klee property in banach and metric spaces in this paper we define a property that implies uniform opial condition or uniform kadec klee property in banach spaces this will unify many examples in the literature finally we will extend these properties into hyperbolic metric spaces

galerkin alternating direction methods for nonrectangular regions using patch approximations galerkin alternating direction procedures are considered for the nonlinear parabolic equation u t sum i j 1 n frac partial partial xi i left tilde a ij xi u frac partial u partial xi j right sum i 1 n tilde b i xi u frac partial u partial xi i f xi t u which is given on a nonrectangular curved region we use isoparametric elements and an approximation to the jacobian of the isoparametric map which is based on patches of finite elements with these procedures a multidimensional problem can be solved as a series of one dimensional problems optimal order l 2 and h 1 error estimates as well as optimal order work estimates are derived

alternating direction implicit galerkin finite element method for the two dimensional fractional diffusion wave equation new numerical techniques are presented for the solution of the two dimensional fractional diffusion wave equation with a time fractional derivative of order 1 2 in these methods galerkin finite element is used for the spatial discretization and for the time stepping new alternating direction implicit adi method based on the crank nicolson method are considered the unconditional stability and l 2 norm convergence of the adi scheme are proved rigorously numerical results are presented to support our theoretical analysis

a mixed volume element with upwind multistep mixed volume element and convergence analysis for numerical simulation of nuclear waste contaminant disposal abstract in this paper the authors discuss a numerical simulation problem of three dimensional contamination treatment from nuclear waste the mathematical model is defined by a nonlinear system of an initial boundary problem of four partial differential equations an elliptic equation for the fluid pressure three convection diffusion equations for the brine radionuclides and heat the pressure appears within the concentration equations and heat conduction equation and the darcy velocity controls the concentrations and the temperature the pressure is solved by the conservative mixed volume element method and the order of accuracy is improved by the darcy velocity the concentrations and temperature are treated by a combination of a multistep method the upwind approximation and a mixed volume element the partial derivatives to time variable are approximated by a multistep method a mixed volume element and an upwind scheme are used to discretize the diffusion and the convection respectively the composite method can solve the convection dominated diffusion problems well because it eliminates numerical dispersion and nonphysical oscillation and has high order computational accuracy the mixed volume element has the local conservation of mass and energy and it can obtain the concentrations and temperature and their adjoint vector functions simultaneously the conservation nature plays an important role in numerical simulation of underground fluid by the technique of a priori estimates of differential equations we derive an optimal second order result in l 2 norm numerical examples are given to show the effectiveness and practicability of our approach and the composite method is testified as a powerful tool for solving the challenging benchmark problem

the piecewise exponential method pem for the numerical simulation of wave propagation a multidimensional wave problem abstract this paper presents the extension of the pem scheme for the case of multidimensional wave problems this is done on the basis of operator splitting and the pem scheme for the one dimensional wave equation given in the previous publication v zalizniak commun numer meth engrg 13 3 1997 163 171 numerical results for two problems of wave propagation in comparison with the analytical solutions are given to demonstrate the effectiveness of the proposed method for elastodynamics

on the convergence of finite difference schemes for parabolic equations with variable coefficients we study the convergence of the finite difference schemes for the first initial boundary value problem for linear second order parabolic equations with variable coefficients using the bilinear version of the bramble hilbert lemma we obtain estimate of convergence in discretew 2 1 1 2 norm compatible with the smoothness of generalized solutionu w 2 2 q 1 3 and coefficients of equation

asymptotic behaviour of quasi autonomous dissipative systems in hilbert spaces abstract by suitably modifying the methods used by b djafari rouhani ergodic theorems for non expansive sequences in hilbert spaces and related problems part i pp 1 76 thesis yale university and a new proof of the weak convergence theorems for non expansive sequences and curves in hilbert spaces preprint 1984 and by b djafari rouhani and s kakutani ergodic theorems for non expansive non linear operators in a hilbert space preprint 1984 we study the asymptotic behaviour of the quasi autonomous dissipative system du dt au where a is a monotone operator in a hilbert space h and l 1 0 h

asymptotic behaviour of almost nonexpansive sequences in a hilbert space abstract the results in 9 11 12 for nonexpansive sequences are generalized to almost nonexpansive sequences in a hilbert space and using this notion a direct proof of a result of h brezis and f e browder 6 is given

a new numerical approach to solve an elliptic equation a new numerical approach to solve an elliptic partial differential equation that originates from the governing equations of steady state fluid flow and heat transfer is presented the elliptic partial differential equation is transformed by introducing an exponential function to eliminate the convection terms in the equation a fourth order central differencing scheme and a second order central differencing scheme are used to numerically solve the transformed elliptic partial differential equation analytical solutions of this equation are also given comparisons are made between the analytical solutions the numerical results using the present schemes and those using the four classical differencing schemes namely the first order upwind scheme hybrid scheme power law scheme and exponential scheme the comparisons illustrate that the proposed algorithm performs better than the four classical differencing schemes

numerical solution of a transformed parabolic equation a numerical analysis of a parabolic partial differential equation pde which originates from the governing equations of transient fluid flow and heat transfer is presented the parabolic pde is transformed by introducing an exponential function to eliminate the convection terms in the equation a fourth order central differencing scheme and a second order central differencing scheme are used to solve the transformed parabolic pde numerically the analytical solutions of this equation are also given comparisons against the analytical solutions are made for the numerical results using the present schemes and those using the four classical differencing schemes namely the first order upwind scheme hybrid scheme power law scheme and exponential scheme

a mean ergodic theorem for nonlinear semigroups which are asymptotically nonexpansive in the intermediate sense abstract let x be a uniformly convex banach space such that its dual x has the kadec klee property c a bounded closed convex subset of x and t t t t 0 a one parameter nonlinear semigroup which is asymptotically nonexpansive in the intermediate sense we show that every continuous almost orbit of t is weakly almost convergent to a common fixed point of t a discrete version of our main result is also included

weak convergence of almost orbits of asymptotically nonexpansive commutative semigroups we give three equivalent conditions for weak convergence of almost orbits of an asymptotically nonexpansive commutative semigroup acting on a nonempty bounded closed convex subset of a uniformly convex banach space whose dual has the kadec property

maximum norm error estimates of efficient difference schemes for second order wave equations the three level explicit scheme is efficient for numerical approximation of the second order wave equations by employing a fourth order accurate scheme to approximate the solution at first time level it is shown that the discrete solution is conditionally convergent in the maximum norm with the convergence order of two since the asymptotic expansion of the difference solution consists of odd powers of the mesh parameters time step and spacings an unusual richardson extrapolation formula is needed in promoting the second order solution to fourth order accuracy extensions of our technique to the classical adi scheme also yield the maximum norm error estimate of the discrete solution and its extrapolation numerical experiments are presented to support our theoretical results

a modified backward time discretization for nonlinear parabolic equations using patch approximations a modified backward difference time discretization is presented for galerkin approximations for the nonlinear parabolic equation in mathbb r d q x u u t sum i j 1 d frac partial partial x i left a ij x u frac partial u partial x j right sum i 1 d b i x u frac partial u partial x i f x t u this procedure uses a local approximation of the coefficients based on patches of finite elements to update the coefficient matrix at each time step and allows for efficient use of such direct methods for solving linear systems of equations as nested dissection optimal order error estimates and almost optimal order work requirements are derived an alternating direction variant is also presented which has both optimal order error estimates and optimal order work requirements

ergodic theorems for hybrid sequences in a hilbert space with applications abstract in this paper we introduce the notion of generalized hybrid sequences extending the notion of nonexpansive sequences introduced and studied in our previous work djafari rouhani 1981 1990 1990 1997 2002 2004 2002 2 3 4 5 6 7 8 and prove ergodic and convergence theorems for such sequences in a hilbert space h subsequently we apply our results to prove new fixed point theorems for generalized hybrid mappings first introduced in kocourek et al 2010 14 takahashi and takeuchi 2011 20 defined on arbitrary nonempty subsets of h

the finite element numerical analysis for a class nonlinear evolution equations a complete stability and convergence analysis is given for two level scheme for the approximate solution of a class generalized nerve conduction equation defining a kind of truncation error and using a kind of technique optimal order estimates in l 2 is constructively obtained

a family of improved falkner type methods for oscillatory systems for solving general second order initial value problems u t f t u t u t the classical falkner methods can date back to falkner s work in 1936 in this paper we propose and study a family of improved falkner type methods for the oscillatory system u t m u t g t u t where g r rd rd in which the first derivative does not appear explicitly and m rd d is a symmetric positive semi definite matrix the new methods take into account the oscillatory structures of the problem and exactly integrate the unperturbed problem u t m u t 0 the global error bounds of the new methods are presented numerical experiments are performed to show that the new methods are more efficient than other effective methods appeared in the scientific literature

an alternating direction implicit backward differentiation orthogonal spline collocation method for linear variable coefficient parabolic equations we present a new fully discrete method to solve a linear variable coefficient parabolic equation on a rectangular region the scheme is an alternating direction implicit method which uses the third order backward differentiation formula for the time discretization and piecewise hermite bicubic orthogonal spline collocation for the spatial discretization the l 2 norm stability and convergence analysis is carried out for the heat equation the stability analysis reveals that the scheme is stable with respect to the right hand sides but it is unstable with respect to the initial conditions we explain using a damping property of the scheme and round off error analysis why this instability is harmless we also show how a different choice of initial values leads to optimal convergence orders numerical results demonstrate the third order accuracy in time and optimal order accuracy in space using maximum and sobolev norms respectively

symmetric finite volume element approximations of second order linear hyperbolic integro differential equations in this paper based on barycenter dual meshes we develop one semi discrete and two full discrete symmetric finite volume element schemes for second order linear hyperbolic integro differential equations the optimal order error estimates in l 2 and h 1 norms are derived for the semi discrete scheme numerical experiments confirm the performance of the symmetric schemes and further show that the l 2 norm convergence rate of the full discrete backward euler and crank nicolson schemes to be o h 2 and o h 2 2 respectively

unconditional superconvergence analysis for nonlinear hyperbolic equation with nonconforming finite element nonlinear hyperbolic equation is studied by developing a linearized galerkin finite element method fem with nonconforming eq1rot element a time discrete system is established to split the error into two parts which are called the temporal error and the spatial error respectively the temporal error is proved skillfully which leads to the analysis for the regularity of the time discrete system the spatial error is derived independently with order o h2 h in broken h1 norm the final unconditional superclose result of u with order o h2 2 is deduced based on the above achievements the two typical characters of this nonconforming eq1rot element see lemma1 below play an important role in the procedure of proof at last a numerical example is provided to support the theoretical analysis here h is the subdivision parameter and the time step

alternating direction multistep methods for parabolic problems iterative stabilization efficient multistep procedures for time stepping galerkin methods for parabolic partial differential equations are presented and analyzed the procedures involve using an alternating direction preconditioned iterative method for approximately solving the linear equations arising at each timestep in a discrete galerkin method optimal order convergence rates are obtained for the iterative methods work estimates of almost optimal order are obtained

boundary and interface conditions within a finite element preconditioner for spectral methods abstract the performances of a finite element preconditioner in the iterative solution of spectral collocation schemes for elliptic boundary value problems is investigated it is shown how to make the preconditioner cheap by adi iterations and how to take advantage of the finite element properties in enforcing neumann and interface conditions in the spectral schemes

two time level adi finite volume method for a class of second order hyperbolic problems in this paper we develop a two time level alternating direction implicit adi method for a class of second order hyperbolic problems on a rectangular domain the method builds on the finite volume method with biquadratic basis functions for the discretization in space and a crank nicolson approach for the time stepping we obtain a second order error estimation in the h 1 norm numerical experiments are performed to demonstrate the theoretical findings

a finite element alternating direction method combined with a modified method of characteristics for convection diffusion problems a finite element alternating direction method is combined with a modified method of characteristics to solve convection diffusion problems this approach combines the attractive attributes of the modified method of characteristics with those of the alternating direction method for convection dominated problems the modified method of characteristics has much smaller time truncation errors than those of standard methods hence larger timesteps can be taken to integrate the convection diffusion equation the alternating direction method reduces multidimensional problems to a series of uncoupled one dimensional problems which results in very low execution times and storage the method has the ability to solve problems on curved structured grids the accuracy of this procedure is demonstrated by linear and nonlinear convection dominated diffusion problems in two space dimensions the procedure can be generalized to more than two dimensions

asymptotic behavior of uniformly asymptotically almost nonexpansive curves in a hilbert space abstract we extend the notion of uniformly asymptotically almost nonexpansive curves which was first introduced in rouhani and kim j nonlinear convex anal 4 2003 175 for a commutative semigroup to a general semitopological semigroup study the asymptotic behavior of such curves in a hilbert space and apply our results to study the asymptotic behavior for almost orbits of semigroups of asymptotically nonexpansive type mappings over a general semitopological semigroup in a hilbert space since in this case the semigroup is not directed the proofs in rouhani and kim j nonlinear convex anal 4 2003 175 do not extend to this case and new methods have to be used for the proofs in addition to rouhani j math anal appl 147 1990 465 j math anal appl 151 1990 226 nonlinear anal 44 2001 627 nonlinear anal 51 2002 735 and rouhani and kim j nonlinear convex anal 4 2003 175 our results extend and unify many previously known results j math anal appl 127 1987 206 proc amer math soc 104 1988 431 nonlinear anal 29 1997 539 topol methods nonlinear anal 5 1995 305 j math anal appl 105 1985 514 nonlinear anal 26 1996 1411 pacific j math 126 1987 277 j math anal appl 206 1997 451 chinese ann math 6 1996 729 nonlinear anal 14 1990 69 j math anal appl 85 1982 172 proc amer math soc 81 1981 253 proc amer math soc 97 1986 55 canad j math 44 1992 880 j math anal appl 142 1989 242 proc amer math soc 117 1993 385

a symmetric characteristic fve method with second order accuracy for nonlinear convection diffusion problems the finite volume element fve methods used currently are essentially low order and unsymmetric in this paper by biquadratic elements and multistep methods we construct a second order fve scheme for nonlinear convection diffusion problem on nonuniform rectangular meshes to overcome the numerical oscillation we discretize the problem along its characteristic direction the choice of alternating direction strategy is critical in this paper which guarantees the high efficiency and symmetry of the discrete scheme optimal order error estimates in h 1 norm are derived and a numerical example is given at the end to confirm the usefulness of the method

galerkin alternating direction methods for a kind of nonlinear hyperbolic equations on nonrectangular regions patch approximation and an approximation to the jacobian of the isoparametric map are applied to discuss a three level and a four level galerkin alternating direction procedures for a kind of nonlinear hyperbolic equation which is given on a nonrectangular curved region with these procedures multi dimensional problems can be solved as a series of one dimensional problems and the l 2 norm error estimates are derived

asymptotic behavior for almost orbits of a reversible semigroup of non lipschitzian mappings in a metric space let m be a metric space and a hausdorff topology on m such that m is compact let s be a right reversible semitopological semigroup and t s s s a representation of s as asymptotically nonexpansive type self mappings of m and u a bounded almost orbit of we study the convergence of the net u s s s in m when the triplet m satisfies various types of opial conditions our results extend and unify many previously known results 2002 elsevier science usa all rights reserved

an alternating direction galerkin method for a class of second order hyperbolic equations in two space variables a new alternating direction implicit adi galerkin method is devised and analyzed for solving a certain class of second order hyperbolic initial boundary value problems in two space variables this class includes the wave equation in cartesian coordinates polar coordinates and cylindrical coordinates with radial symmetry optimal a priori h 0 1 and l 2 error estimates are derived

existence and approximation of fixed points of firmly nonexpansive type mappings in banach spaces a class of nonlinear operators in banach spaces is proposed we call each operator in this class a firmly nonexpansive type mapping this class contains the classes of firmly nonexpansive mappings in hilbert spaces and resolvents of maximal monotone operators in banach spaces we study the existence and approximation of fixed points of firmly nonexpansive type mappings in banach spaces

a new numerical algorithm based on transformed equations and its applications to very low re fluid flows a new numerical algorithm is presented in this study to solve the partial differential equations that govern fluid flows this new algorithm is based on first transforming the partial differential equations by introducing an exponential function to eliminate the convection terms a fourth order central differencing scheme and a second order central differencing scheme are used to discretize the transformed equations the algorithm is then applied to simulate fluid flows with exact solutions to validate this new algorithm the fluid flows used in this study are a self designed quasi fluid flow problem stagnation in plane flow hiemenz flow and flow between two concentric cylinders comparisons against the exact solution are made for the results obtained using the new numerical algorithm as well as the power law scheme the comparisons indicate that the present fourth order scheme performs the best and the present second order scheme is the next most accurate