elliptic basis for the zernike system heun function solutions the differential equation that defines the zernike system originally proposed to classify wavefront aberrations of the wavefields in the disk of a circular pupil had been shown to separate in three distinct coordinate systems obtained from polar coordinates on a half sphere here we find and examine the separation in the generic elliptic coordinate system on the half sphere and its projected disk where the solutions separated in jacobi coordinates contain heun polynomials

a modified weak galerkin finite element method for a class of parabolic problems in this paper we consider the solution of parabolic equation using the modified weak galerkin finite element procedure which is named as mwg fem based on the conception of the modified weak derivative over discontinuous functions with heterogeneous properties in which the classical gradient operator is replaced by a modified weak gradient operator optimal order error estimates in a discrete l2 norm and h1 norm are established for the corresponding modified weak galerkin finite element solutions finally we numerically verify the convergence theory for the mwg fem through some examples

unified hybridization of discontinuous galerkin mixed and continuous galerkin methods for second order elliptic problems we introduce a unifying framework for hybridization of finite element methods for second order elliptic problems the methods fitting in the framework are a general class of mixed dual finite element methods including hybridized mixed continuous galerkin nonconforming and a new wide class of hybridizable discontinuous galerkin methods the distinctive feature of the methods in this framework is that the only globally coupled degrees of freedom are those of an approximation of the solution defined only on the boundaries of the elements since the associated matrix is sparse symmetric and positive definite these methods can be efficiently implemented moreover the framework allows in a single implementation the use of different methods in different elements or subdomains of the computational domain which are then automatically coupled finally the framework brings about a new point of view thanks to which it is possible to see how to devise novel methods displaying very localized and simple mortaring techniques as well as methods permitting an even further reduction of the number of globally coupled degrees of freedom

interbasis expansions in the zernike system the differential equation with free boundary conditions on the unit disk that was proposed by frits zernike in 1934 to find jacobi polynomial solutions indicated as i serves to define a classical system and a quantum system which have been found to be superintegrable we have determined two new orthogonal polynomial solutions indicated as ii and iii that are separable and involve legendre and gegenbauer polynomials here we report on their three interbasis expansion coefficients between the i ii and i iii bases they are given by f23 1 polynomials that are also special su 2 clebsch gordan coefficients and hahn polynomials between the ii iii bases we find an expansion expressed by f34 1 s and racah polynomials that are related to the wigner 6j coefficients

superintegrable classical zernike system we consider the differential equation that zernike proposed to classify aberrations of wavefronts in a circular pupil as if it were a classical hamiltonian with a non standard potential the trajectories turn out to be closed ellipses we show that this is due to the existence of higher order invariants that close into a cubic higgs algebra the zernike classical system thus belongs to the class of superintegrable systems its hamilton jacobi action separates into three vertical projections of polar coordinates of sphere polar and equidistant coordinates on half hyperboloids and also in elliptic coordinates on the sphere

weak galerkin method for the biot s consolidation model abstract we develop a weak galerkin wg finite element method for the biot s consolidation model in the classical displacement pressure two field formulation weak galerkin linear finite elements are used for both displacement and pressure approximations in spatial discretizations backward euler scheme is used for temporal discretization in order to obtain an implicit fully discretized scheme we study the well posedness of the linear system at each time step and also derive the overall optimal order convergence of the wg formulation such wg scheme is designed on general shape regular polytopal meshes and provides stable and oscillation free approximation for the pressure without special treatment numerical experiments are presented to demonstrate the efficiency and accuracy of the proposed weak galerkin finite element method

a modified weak galerkin finite element method in this paper we introduce a new discrete weak gradient operator and a new weak galerkin wg finite element method for second order poisson equations based on this new operator this newly defined discrete weak gradient operator allows us to use a single stabilizer which is similar to the one used in the discontinuous galerkin dg methods without having to worry about choosing a sufficiently large parameter in addition we will establish the optimal convergence rates and validate the results with numerical examples

a globally convergent version of the polak ribiere conjugate gradient method in this paper we propose a new line search algorithm that ensures global convergence of the polak ribiere conjugate gradient method for the unconstrained minimization of nonconvex differentiable functions in particular we show that with this line search every limit point produced by the polak ribiere iteration is a stationary point of the objective function moreover we define adaptive rules for the choice of the parameters in a way that the first stationary point along a search direction can be eventually accepted when the algorithm is converging to a minimum point with positive definite hessian matrix under strong convexity assumptions the known global convergence results can be reobtained as a special case from a computational point of view we may expect that an algorithm incorporating the step size acceptance rules proposed here will retain the same good features of the polak ribiere method while avoiding pathological situations

a nonlinear conjugate gradient method with a strong global convergence property conjugate gradient methods are widely used for unconstrained optimization especially large scale problems the strong wolfe conditions are usually used in the analyses and implementations of conjugate gradient methods this paper presents a new version of the conjugate gradient method which converges globally provided the line search satisfies the standard wolfe conditions the conditions on the objective function are also weak being similar to those required by the zoutendijk condition

an a posteriori error estimator for the weak galerkin least squares finite element method abstract in this paper we derive an a posteriori error estimator for the weak galerkin least squares wg ls method applied to the reaction diffusion equation we show that this estimator is both reliable and efficient allowing it to be used for adaptive refinement due to the flexibility of the wg ls discretization we are able to design a simple and straightforward refinement scheme that is applicable to any shape regular polygonal mesh finally we present numerical experiments that confirm the effectiveness of the estimator and demonstrate the robustness and efficiency of the proposed adaptive wg ls approach

a weak galerkin finite element method for a coupled stokes darcy problem on general meshes abstract in this paper we introduce and analyze a weak galerkin finite element method for numerically solving the coupling of fluid flow with porous media flow flows are governed by the stokes equations in primal velocity pressure formulation and darcy equation in the second order primary formulation respectively and the corresponding transmission conditions are given by mass conservation balance of normal forces and the beavers joseph saffman law by using the weak galerkin approach we consider the two dimensional problem with the usual polynomials of degree k 1 for the velocity and hydraulic head while polynomials of degree k 1 for the pressure the velocity and hydraulic head is enhanced by polynomials of degree k 1 on the edge of a finite element partition this new method has a lot of attractive computational features more general finite element partitions of arbitrary polygons with certain shape regularity fewer degrees of freedom and parameter free stability and error estimates of optimal order are obtained moreover numerical experiences are presented to illustrate the good performance confirm the optimal order of convergence and verify the efficiency of the proposed weak galerkin method in this paper we also deal with finite element partitions consisting of general meshes such as triangular mesh quadrilateral mesh hexagonal dominant mesh and voronoi mesh for the numerical weak galerkin approximation

an analysis of a weak galerkin finite element method for stationary navier stokes problems abstract in this article we present and analyze a weak galerkin finite element method for stationary navier stokes problems this weak galerkin finite element scheme is based on a shape regular partition consisting of arbitrary polygons polyhedra we first establish a discrete embedding inequality that is useful in weak galerkin finite element analysis for nonlinear problems then the stability and unique existence are proved for the discrete velocity and pressure by means of a discrete inf sup condition furthermore we derive the optimal error estimates for velocity approximation in the discrete h 1 norm and pressure approximation in the l 2 norm respectively numerical examples are provided that corroborate the optimal convergence of the proposed method

a note about wyl s conjugate gradient method and its applications this paper reviews the development of different versions of nonlinear conjugate gradient methods shortly the special attentions were given to the wyl method which was proposed by zengxin wei et al the convergence properties of some conjugate gradient methods applied mathematics and computation 183 2006 1341 1350 and its applications

the proof of the sufficient descent condition of the wei yao liu conjugate gradient method under the strong wolfe powell line search abstract in z wei et al the convergence properties of some conjugate gradient methods appl math comput 183 2006 1341 1350 wei et al proposed a new conjugate gradient method called wyl method which has good numerical experiments under the sufficient descent condition and wwp line search the given method s convergent results have been established by wei et al so the sufficient descent condition is important for the wyl method in this paper we prove that by restricting the parameter 1 4 under the swp line search condition the wyl method possess the sufficient descent condition which deduces the global convergent results of the wyl method under swp line search

completeness of superintegrability in two dimensional constant curvature spaces we classify the hamiltonians h px2 py2 v x y of all classical superintegrable systems in two dimensional complex euclidean space with two additional second order constants of the motion we similarly classify the superintegrable hamiltonians h j12 j22 j32 v x y z on the complex two sphere where x2 y2 z2 1 this is achieved in all generality using properties of the complex euclidean group and the complex orthogonal group

a new modified weak galerkin finite element scheme for solving the stationary stokes equations abstract in this paper a modified weak galerkin method is proposed for the stokes problem the numerical scheme is based on a novel variational form of the stokes problem the degree of freedoms in the modified weak galerkin method is less than that in the original weak galerkin method while the accuracy stays the same in this paper the optimal convergence orders are given and some numerical experiments are presented to verify the theory

zernike functions rigged hilbert spaces and potential applications we revise the symmetries of the zernike polynomials that determine the lie algebra su 1 1 su 1 1 we show how they induce discrete as well as continuous bases that coexist in the framework of rigged hilbert spaces we also discuss some other interesting properties of zernike polynomials and zernike functions one of the areas of interest of zernike functions has been their applications in optics here we suggest that operators on the spaces of zernike functions may play a role in optical image processing we revise the symmetries of the zernike polynomials that determine the lie algebra su 1 1 su 1 1 we show how they induce discrete as well as continuous bases that coexist in the framework of rigged hilbert spaces we also discuss some other interesting properties of zernike polynomials and zernike functions one of the areas of interest of zernike functions has been their applications in optics here we suggest that operators on the spaces of zernike functions may play a role in optical image processing

a weak galerkin finite element method for the navier stokes equations abstract this paper introduces a weak galerkin wg finite element method for the navier stokes equations in the primal velocity pressure formulation optimal order error estimates are established for the corresponding numerical approximations it must be emphasized that the wg finite element method is designed on finite element partitions consisting of arbitrary shape of polygons or polyhedra which are shape regular numerical experiments are presented to support the theoretical results

weak galerkin methods for second order elliptic interface problems weak galerkin methods refer to general finite element methods for partial differential equations pdes in which differential operators are approximated by their weak forms as distributions such weak forms give rise to desirable flexibilities in enforcing boundary and interface conditions a weak galerkin finite element method wg fem is developed in this paper for solving elliptic pdes with discontinuous coefficients and interfaces theoretically it is proved that high order numerical schemes can be designed by using the wg fem with polynomials of high order on each element extensive numerical experiments have been carried out to validate the wg fem for solving second order elliptic interface problems high order of convergence is numerically confirmed in both l 2 and l norms for the piecewise linear wg fem special attention is paid to solve many interface problems in which the solution possesses a certain singularity due to the nonsmoothness of the interface a challenge in research is to design nearly second order numerical methods that work well for problems with low regularity in the solution the best known numerical scheme in the literature is of order o h to o h 1 5 for the solution itself in l norm it is demonstrated that the wg fem of the lowest order i e the piecewise constant wg fem is capable of delivering numerical approximations that are of order o h 1 75 to o h 2 in the l norm for c 1 or lipschitz continuous interfaces associated with a c 1 or h 2 continuous solution

a weak galerkin finite element method for burgers equation abstract we propose a weak galerkin wg finite element method for solving the one dimensional burgers equation based on a new weak variational form both semi discrete and fully discrete wg finite element schemes are established and analyzed we prove the existence of the discrete solution and derive the optimal order error estimates in a discrete h 1 norm and the standard l 2 norm respectively numerical experiments are presented to illustrate our theoretical analysis

global convergence properties of conjugate gradient methods for optimization this paper explores the convergence of nonlinear conjugate gradient methods without restarts and with practical line searches the analysis covers two classes of methods that are globally convergent on smooth nonconvex functions some properties of the fletcher reeves method play an important role in the first family whereas the second family shares an important property with the polak ribiere method numerical experiments are presented

the weak galerkin finite element method for the transport reaction equation abstract we present and analyze a weak galerkin finite element method for solving the transport reaction equation in d space dimensions this method is highly flexible by allowing the use of discontinuous finite element on general meshes consisting of arbitrary polygon polyhedra we derive the l 2 error estimate of o h k 1 2 order for the discrete solution when the kth order polynomials are used for k 0 moreover for a special class of meshes we also obtain the optimal error estimate of o h k 1 order in the l 2 norm a derivative recovery formula is presented to approximate the convection directional derivative and the corresponding superconvergence estimate is given numerical examples on compatible and non compatible meshes are provided to show the effectiveness of this weak galerkin method

global convergence of a two parameter family of conjugate gradient methods without line search we study the global convergence of a two parameter family of conjugate gradient methods in which the line search procedure is replaced by a fixed formula of stepsize this character is of significance if the line search is expensive in a particular application in addition to the convergence results we present computational results for various conjugate gradient methods without line search including those discussed by sun and zhang ann oper res 103 2001 161 173

the convergence properties of some new conjugate gradient methods abstract in this paper a new conjugate gradient formula k is given to compute the search directions for unconstrained optimization problems general convergence results for the proposed formula with some line searches such as the exact line search the wolfe powell line search and the grippo lucidi line search are discussed under the above line searches and some assumptions the global convergence properties of the given methods are discussed the given formula k 0 and has the similar form with k prp preliminary numerical results show that the proposed methods are efficient

a discontinuous skeletal method for the viscosity dependent stokes problem we devise and analyze arbitrary order nonconforming methods for the discretization of the viscosity dependent stokes equations on simplicial meshes we keep track explicitly of the viscosity and aim at pressure robust schemes that can deal with the practically relevant case of body forces with large curl free part in a way that the discrete velocity error is not spoiled by large pressures the method is inspired from the recent hybrid high order hho methods for linear elasticity after elimination of the auxiliary variables by static condensation the linear system to be solved involves only discrete face based velocities which are polynomials of degree k ge 0 and cell wise constant pressures our main result is a pressure independent energy error estimate on the velocity of order k 1 the main ingredient to achieve pressure independence is the use of a divergence preserving velocity reconstruction operator in the discretization of the body forces we also prove an l 2 pressure estimate of order k 1 and an l 2 velocity estimate of order k 2 the latter under elliptic regularity the local mass and momentum conservation properties of the discretization are also established finally two and three dimensional numerical results are presented to support the analysis

an effective implementation for stokes equation by the weak galerkin finite element method abstract in this paper we introduce and analyze the schur complement technique to a weak galerkin wg for short finite element method for solving stokes equation due to the special structure of the weak functions the main idea of schur decomposition method is to express the interior velocity functions defined inside partition elements in terms of the velocity functions defined on the partition boundary and the pressure functions this can largely reduce the degree of freedom of the derived discrete linear system we investigate the well posedness of the solution generated from schur complement and its coincidence with the standard wg method finally we exhibit some numerical experiments to illustrate the theoretical analysis

some modified conjugate gradient methods for unconstrained optimization conjugate gradient methods are highly useful for solving large scale optimization problems because they do not require the storage of any matrices motivated by the construction of conjugate gradient parameters in some existing conjugate gradient methods we propose four modified conjugate gradient methods named nvls nvprp nvhs and nvls respectively and prove that these methods with the strong wolfe line search possess sufficient descent property and are globally convergent when the parameter in line search conditions is restricted in some suitable interval preliminary numerical results show that the nvprp nvhs and nvls methods are more efficient than many existing conjugate gradient methods for a large number of test problems from a cuter collection

quantum superintegrable zernike system we consider the differential equation that zernike proposed to classify aberrations of wavefronts in a circular pupil whose value at the boundary can be nonzero on this account the quantum zernike system where that differential equation is seen as a schrodinger equation with a potential is special in that it has a potential and a boundary condition that are not standard in quantum mechanics we project the disk on a half sphere and there we find that in addition to polar coordinates this system separates into two additional coordinate systems non orthogonal on the pupil disk which lead to schrodinger type equations with poschl teller potentials whose eigen solutions involve legendre gegenbauer and jacobi polynomials this provides new expressions for separated polynomial solutions of the original zernike system that are real the operators which provide the separation constants are found to participate in a superintegrable cubic higgs algebra

weak galerkin finite element method for viscoelastic wave equations abstract in this article we consider a weak galerkin finite element method wg fem for solving one type of viscoelastic wave equation a discrete weak gradient operator on discontinuous piecewise polynomials is used in the numerical scheme we introduce both the semi and fully discrete wg fems and establish the corresponding stability estimates optimal order error estimate in h 1 norm is derived numerical experiments are preformed to verify the theoretical results

convergence properties of nonlinear conjugate gradient methods recently important contributions on convergence studies of conjugate gradient methods were made by gilbert and nocedal siam j optim 2 1992 pp 21 42 they introduce a sufficient descent condition to establish global convergence results although this condition is not needed in the convergence analyses of newton and quasi newton methods gilbert and nocedal hint that the sufficient descent condition which was enforced by their two stage line search algorithm may be crucial for ensuring the global convergence of conjugate gradient methods this paper shows that the sufficient descent condition is actually not needed in the convergence analyses of conjugate gradient methods consequently convergence results on the fletcher reeves and polak ribiere type methods are established in the absence of the sufficient descent condition to show the differences between the convergence properties of fletcher reeves and polak ribiere type methods two examples are constructed showing that neither the boundedness of the level set nor the restriction beta k geq 0 can be relaxed for the polak ribiere type methods

the weak galerkin finite element method for the symmetric hyperbolic systems abstract in this paper we present and analyze a weak galerkin finite element wg method for solving the symmetric hyperbolic systems this method is highly flexible by allowing the use of discontinuous finite elements on element and its boundary independently of each other by introducing special weak derivative we construct a stable weak galerkin scheme and derive the optimal l 2 error estimate of o h k 1 2 order for the discrete solution when the k order polynomials are used for k 0 as application we discuss this wg method for solving the singularly perturbed convection diffusion reaction equation and derive an e uniform error estimate of order k 1 2 numerical examples are provided to show the effectiveness of the proposed wg method

fourth order superintegrable systems separating in cartesian coordinates i exotic quantum potentials a study is presented of two dimensional superintegrable systems separating in cartesian coordinates and allowing an integral of motion that is a fourth order polynomial in the momenta all quantum mechanical potentials that do not satisfy any linear differential equation are found they do however satisfy nonlinear odes we show that these equations always have the painleve property and integrate them in terms of known painleve transcendents or elliptic functions

the weak galerkin finite element method for incompressible flow abstract we study the weak galerkin finite element method for stationary navier stokes problem we propose a weak finite element velocity pressure space pair that satisfies the discrete inf sup condition this space pair is then employed to construct a stable weak galerkin finite element scheme without adding any stabilizing term or penalty term we prove a discrete embedding inequality on the weak finite element space which together with the discrete inf sup condition enables us to establish the unique existence and stability estimates of the discrete velocity and pressure then we derive the optimal error estimates for velocity and pressure approximations in the h 1 norm and l 2 norm respectively numerical experiments are provided to illustrate the theoretical analysis

superconvergence of weak galerkin finite element approximation for second order elliptic problems by l2 projections the weak galerkin finite element method wg fem is a novel numerical method that was first proposed and analyzed by wang and ye 2013 12 for general second order elliptic problems on triangular meshes based on a discrete weak gradient in general the weak galerkin finite element formulations for partial differential equations can be derived naturally by replacing usual derivatives by weakly defined derivatives in the corresponding variational forms the superconvergence in the finite element method is a phenomenon in which the finite element approximation converges to the exact solution at a rate higher than the optimal order error estimate wang 2000 11 proposed and analyzed superconvergence of the standard galerkin finite element method by l 2 projections the main idea behind the l 2 projections is to project the finite element solution to another finite element space with a coarse mesh and a higher order of polynomials the objective of this paper is to establish a general superconvergence result for the weak galerkin finite element approximations for second order elliptic problem by l 2 projection methods the results of numerical experiments show great promise for the robustness reliability flexibility and accuracy of superconvergence in wg fem by l 2 projection methods

a weak galerkin mixed finite element method for second order elliptic problems abstract a new weak galerkin wg method is introduced and analyzed for the second order elliptic equation formulated as a system of two first order linear equations this method called wg mfem is designed by using discontinuous piecewise polynomials on finite element partitions with arbitrary shape of polygons polyhedra the wg mfem is capable of providing very accurate numerical approximations for both the primary and flux variables allowing the use of discontinuous approximating functions on arbitrary shape of polygons polyhedra makes the method highly flexible in practical computation optimal order error estimates in both discrete h and l norms are established for the corresponding weak galerkin mixed finite element solutions

a stable weak galerkin finite element method for stokes problem abstract we study the weak galerkin finite element method for stokes problem a new weak galerkin finite element velocity pressure space pair is presented which satisfies the discrete inf sup condition based on this space pair we establish a stable weak galerkin approximation scheme without adding any stability term or penalty term then we further derive the optimal error estimates for velocity and pressure approximations respectively numerical experiments are provided to illustrate the theoretical analysis

a new weak galerkin finite element scheme for general second order elliptic problems abstract a new weak galerkin wg finite element scheme is presented for general second order elliptic problems in this paper in this new scheme a skew symmetric form has been used for handling the convection term the advantage of the new scheme is that the system of linear equations from the scheme is positive definite and one might easily get the well posedness of the system in this scheme the wg elements are designed to have the form of p k t p k 1 e that is we choose the polynomials of degree k 1 on each element and the polynomials of degree k 1 on the edge face of each element as a result fewer degrees of freedom are generated in the new wg finite element scheme it is also worth pointing out that the wg finite element scheme is established on finite element partitions consisting of arbitrary shape of polygons polyhedra which are shape regular optimal order error estimates are presented for the corresponding numerical approximation in various norms some numerical results are reported to confirm the theory

a weak finite element method for elliptic problems in one space dimension we present a weak finite element method for elliptic problems in one space dimension our analysis shows that this method has more advantages than the known weak galerkin method proposed for multi dimensional problems for example it has higher accuracy and the derived discrete equations can be solved locally element by element we derive the optimal error estimates in the discrete h1 norm the l2 norm and l norm respectively moreover some superconvergence results are also given finally numerical examples are provided to illustrate our theoretical analysis

another improved wei yao liu nonlinear conjugate gradient method with sufficient descent property abstract recently zhang 13 take a little modification to the wei yao liu nonlinear conjugate gradient method proposed by wei et al 10 such that the modified method called nprp method in zhang 13 satisfies sufficient descent condition with greater parameter 0 1 2 in the strong wolfe line search and converges globally for nonconvex minimization with the strong wolfe line search in this paper we take a little modification to the nprp method such that the modified nprp method possesses better properties than the nprp method in zhang 13 firstly the modified nprp method possesses the sufficient descent property without any line searches secondly the modified nprp method converges globally for nonconvex minimization with wolfe line search or armijo line search moreover we extend these results to the hestenes stiefel hs method and prove that the modified hs method also possesses sufficient descent property and global convergence with the standard wolfe conditions numerical results are reported by utilizing some test problems in the cute library

a weak galerkin finite element method for second order elliptic problems this paper introduces a finite element method by using a weakly defined gradient operator over generalized functions the use of weak gradients and their approximations results in a new concept called discrete weak gradients which is expected to play an important role in numerical methods for partial differential equations this article intends to provide a general framework for managing differential operators on generalized functions as a demonstrative example the discrete weak gradient operator is employed as a building block in the design of numerical schemes for a second order elliptic problem in which the classical gradient operator is replaced by the discrete weak gradient the resulting numerical scheme is called a weak galerkin wg finite element method it can be seen that the weak galerkin method allows the use of totally discontinuous functions in the finite element procedure for the second order elliptic problem an optimal order error estimate in both a discrete h 1 and l 2 norms are established for the corresponding weak galerkin finite element solutions a superconvergence is also observed for the weak galerkin approximation

spherical geometry zernike s separability and interbasis expansion coefficients free motion on a 3 sphere properly projected on the 2 dimensional manifold of a disk yields the zernike system which exhibits the fundamental properties of superintegrability these include separability in a variety of coordinate systems polynomial solutions and a particular subset of clebsch gordan coefficients as interbasis expansion coefficients that are higher orthogonal polynomials from the askey scheme deriving these results from the initial formulation in spherical geometry provides the zernike system with interest beyond its optical applications

superconvergence of a modified weak galerkin approximation for second order elliptic problems by l2 projection method abstract this paper derives a superconvergence result for the modified weak galerkin mwg finite element method of the second order elliptic problem the convergence rate of the mwg approximation is improved by 30 after applying a low cost l 2 projection post processing technique these superconvergence phenomena are proved theoretically and confirmed numerically

split least squares finite element methods for linear and nonlinear parabolic problems in this paper we propose some least squares finite element procedures for linear and nonlinear parabolic equations based on first order systems by selecting the least squares functional properly each proposed procedure can be split into two independent symmetric positive definite sub procedures one of which is for the primary unknown variable u and the other is for the expanded flux unknown variable s optimal order error estimates are developed finally we give some numerical examples which are in good agreement with the theoretical analysis

a modified wei yao liu conjugate gradient method for unconstrained optimization in this paper we give a modified wei yao liu conjugate gradient method wei et al 2006 18 which will reduce to the original wei yao liu method and possess the sufficient descent property without any line search furthermore we prove that the presented method is globally convergent for nonconvex functions with the weak wolfe powell line search in a similar way we also extend these results to the modified liu storey method preliminary numerical results show that the proposed methods are effective for the given test problems

some global convergence properties of the wei yao liu conjugate gradient method with inexact line search abstract in z wei s yao l liu the convergence properties of some new conjugate gradient methods applied mathematics and computation 183 2006 1341 1350 wei et al proposed a new conjugate gradient method called wyl method which has good numerical experiments and some excellent properties such as k wyl 0 in this paper we prove that while t k 1 c 2 l g k 2 d k 2 the sufficient descent condition can be satisfied at each iteration of the wyl method based on this the global convergence of the wyl method with the standard armijo line search and the generalized wolfe powell line search is established

a modified weak galerkin finite element method for the stokes equations a modified weak galerkin mwg finite element method is introduced for the stokes equations in the primary velocity pressure formulation this method is based on the weak galerkin wg finite element method developed recently in wang and ye 0000 the unknowns associated with element boundaries in the wg method have been eliminated in the mwg method as the result the mwg method has fewer degrees of freedom optimal order error estimates are established for the corresponding numerical approximation in various norms numerical results are presented to confirm the theory

the clebsch gordan coefficients of the three dimensional lorentz algebra in the parabolic basis starting from the oscillator representation of the three dimensional lorentz algebra so 2 1 we build a lie algebra of second order differential operators which realizes all series of self adjoint irreducible representations the choice of the common self adjoint extention over a two chart function space determines whether they lead to single or multivalued representations over the corresponding lie group the diagonal operator defining the basis is the parabolic subgroup generator the direct product of two such algebras allows for the calculation of all clebsch gordan coefficients explicitly as solutions of schrodinger equations for poschl teller potentials over one d d two d c or three c c charts all coefficients are given in terms of up to two 2f1 hypergeometric functions

generalized zernike or disc polynomials we investigate generalized zernike or disc polynomials p m n z z which are orthogonal 2d polynomials in the unit disc 0 zz 1 is a free parameter these polynomials can be expressed by jacobi polynomials of transformed arguments in connection with a simple angle dependence a limiting procedure leads to laguerre 2d polynomials lm n z z furthermore we introduce the corresponding orthonormalized disc functions the disc polynomials and disc functions obey two differential equations a first order and a second order one with a certain degree of freedom and the operators of lowering and raising of the indices are found these operators can be closed to a lie algebra su 1 1 su 1 1 new generating functions are derived from an operational representation which is alternative to the rodrigues type representation the one dimensional analogue of the disc polynomials which are orthogonal polynomials in the interval 0 r 1 with weight factors 1 r2 x are ultraspherical or gegenbauer polynomials in a new standardization the lowering and raising operators to the corresponding orthonormalized functions form a simple su 1 1 lie algebra this is given in the appendix in sketched form

superintegrability and associated polynomial solutions euclidean space and the sphere in two dimensions in this work we examine the basis functions for those classical and quantum mechanical systems in two dimensions which admit separation of variables in at least two coordinate systems we do this for the corresponding systems defined in euclidean space and on the two dimensional sphere we present all of these cases from a unified point of view in particular all of the special functions that arise via variable separation have their essential features expressed in terms of their zeros the principal new results are the details of the polynomial bases for each of the nonsubgroup bases not just the subgroup cartesian and polar coordinate cases and the details of the structure of the quadratic algebras we also study the polynomial eigenfunctions in elliptic coordinates of the n dimensional isotropic quantum oscillator

a coupling of weak galerkin and mixed finite element methods for poroelasticity abstract in this paper we present a coupling of a weak galerkin method for the displacement of the solid phase with a standard mixed finite method for the pressure and velocity of the fluid phase in poroelasticity equation because our method provides a stable element combination for the mixed linear elasticity problem it can avoid the poroelasticity locking mathematically indeed uniform in time error estimates of all the unknowns are obtained for both semidiscrete scheme and fully discrete scheme without assuming that the constrained specific storage coefficient is uniformly positive compared with the method proposed by jeonghun j lee our method can get the arbitrary convergence order by changing degree of the polynomial as for the computational cost we can use the modified weak galerkin method to eliminate the boundary term and reduce the number of unknowns in addition we do not use gronwall inequality in the error analysis finally the numerical experiments verify the theoretical analysis and show the effectiveness to overcome spurious pressure oscillations

an h 1 galerkin mixed finite element method for parabolic partial differential equations in this paper an h1 galerkin mixed finite element method is proposed and analyzed for parabolic partial differential equations with nonselfadjoint elliptic parts compared to the standard h1 galerkin procedure c1 continuity for the approximating finite dimensional subspaces can be relaxed for the proposed method moreover it is shown that the finite element approximations have the same rates of convergence as in the classical mixed method but without lbb consistency condition and quasiuniformity requirement on the finite element mesh finally a better rate of convergence for the flux in l2 norm is derived using a modified h1 galerkin mixed method in two and three space dimensions which confirms the findings in a single space variable and also improves upon the order of convergence of the classical mixed method under extra regularity assumptions on the exact solution

a weak galerkin finite element method for solving nonlinear convection diffusion problems in two dimensions abstract we study weak galerkin wg finite element method fem for solving nonlinear convection diffusion problems a wg finite element scheme is presented based on a new variational form we prove the energy conservation law and stability of the continuous time wg fem in particular optimal order error estimates are established for the wg fem approximation in both a discrete h1 norm and l2 norm numerical experiments are performed to confirm the theoretical results

an improved wei yao liu nonlinear conjugate gradient method for optimization computation in this paper we take a little modification to the wei yao liu nonlinear conjugate gradient method proposed by wei et al z wei s yao l liu the convergence properties of some new conjugate gradient methods appl math comput 183 2006 1341 1350 such that the modified method possesses better convergence properties in fact we prove that the modified method satisfies sufficient descent condition with greater parameter email protected 0 12 in the strong wolfe line search and converges globally for nonconvex minimization we also extend these results to the hestenes stiefel method and prove that the modified hs method is globally convergent for nonconvex functions with the standard wolfe conditions numerical results are reported by using some test problems in the cute library

a class of one parameter conjugate gradient methods this paper proposes a class of one parameter conjugate gradient methods which can be regarded as some kinds of convex combinations of some modified form of prp and hs methods the scalar k has the form of k k 1 k the convergence of the given methods is analyzed by some unified tools which show the global convergence of the proposed methods numerical experiments with the cute collections show that the proposed methods are promising

least squares galerkin methods for parabolic problems i semidiscretization in time this is the first part of a series of papers on least squares galerkin methods for parabolic initial boundary value problems these methods are based on the minimization of a least squares functional for an equivalent first order system over space and time with respect to suitable discrete spaces this paper presents the derivation and analysis of one step methods for semi discretization in time from least squares principles for linear parabolic problems one of the most important features of the least squares methodology is the built in a posteriori estimate for the approximation error this is a consequence of the equivalence of the least squares functional to the consistency error associated with a time step measured in an appropriate norm for the presentation in this paper we focus our attention on the specific combination of piecewise linear not necessarily continuous functions with continuous piecewise linear functions for the flux and scalar variable respectively for the resulting method a convergence result is shown for the scalar variable and the associated flux