a projection method for constructing a mass conservative velocity field abstract in the numerical modeling of fluid flow and transport problems the velocity field frequently needs to be projected from one finite dimensional space into another in certain applications especially those involving modeling of multi species transport the new projected velocity field should be accurate as well as locally mass conservative in this paper a velocity projection method has been developed that is both accurate and mass conservative element by element on the projected grid the velocity correction is expressed as gradient of a scalar pressure field and the resultant poisson equation is solved using a mixed hybrid finite element method and lowest order raviart thomas spaces the conservative projection method is applied to the system of shallow water equations and a theoretical error estimate is derived

reliable computation of the zeros of solutions of second order linear odes using a fourth order method a fourth order fixed point method to compute the zeros of solutions of second order homogeneous linear odes is obtained from the approximate integration of the riccati equation associated with the ode the method requires the evaluation of the logarithmic derivative of the function and also uses the coefficients of the ode an algorithm to compute with certainty all the zeros in an interval is given which provides a fast reliable and accurate method of computation the method is illustrated by the computation of the zeros of gauss hypergeometric functions including jacobi polynomials and confluent hypergeometric functions laguerre polynomials hermite polynomials and bessel functions included among others the examples show that typically 4 or 5 iterations per root are enough to provide more than 100 digits of accuracy without requiring a priori estimations of the roots

analytical and numerical aspects of a generalization of the complementary error function in this paper we discuss analytical and numerical properties of the function v z r 1 0 e zt t t dt with z 0 which can be viewed as a generalization of the complementary error function and in fact also as a generalization of the kummer u function the function v z is used for certain values of the parameters as an approximant in a singular perturbation problem we consider the relation with other special functions and give asymptotic expansions as well as recurrence relations several methods for its numerical evaluation and examples are given 2000 mathematics subject classification 33b20 33c15 41a60 65d20 65q05

computing the hypergeometric function the hypergeometric function of a real variable is computed for arbitrary real parameters the transformation theory of the hypergeometric function is used to obtain rapidly convergent power series the divergences that occur in the individual terms of the transformation for integer parameters are removed using a finite difference technique

nil clean matrix rings abstract we characterize the nil clean matrix rings over fields as a by product we obtain a complete characterization of the finite rank abelian groups with nil clean endomorphism ring and the abelian groups with strongly nil clean endomorphism ring respectively

when is every linear transformation a sum of two commuting invertible ones abstract a well known result of wolfson 7 and zelinsky 8 says that every linear transformation of a vector space v over a division ring d is a sum of two invertible linear transformations except when dim v 1 and d f 2 indeed many of these linear transformations satisfy a stronger property that they are sums of two commuting invertible linear transformations the goal of this note is to prove that every linear transformation of a vector space v over a division ring d is a sum of two commuting invertible ones if and only if d 3 and dim v as a consequence a sufficient and necessary condition is obtained for a semisimple module to have the property that every endomorphism is a sum of two commuting automorphisms

numerically satisfactory solutions of hypergeometric recursions each family of gauss hypergeometric functions fn 2f1 a 1n b 2n c 3n z n z for fixed j 0 1 not all j equal to zero satisfies a second order linear difference equation of the form anfn 1 bnfn cnfn 1 0 because of symmetry relations and functional relations for the gauss functions many of the 26 cases for different j values can be transformed into each other in this way only with four basic difference equations all other cases can be obtained for each of these recurrences we give pairs of numerically satisfactory solutions in the regions in the complex plane where t1 6 t2 t1 and t2 being the roots of the characteristic equation

when is every matrix over a division ring a sum of an idempotent and a nilpotent abstract a ring is called nil clean if each of its elements is a sum of an idempotent and a nilpotent in response to a question of s breaz et al in 1 we prove that the n n matrix ring over a division ring d is a nil clean ring if and only if d f 2 as consequences it is shown that the n n matrix ring over a strongly regular ring r is a nil clean ring if and only if r is a boolean ring and that a semilocal ring r is nil clean if and only if its jacobson radical j r is nil and r j r is a direct product of matrix rings over f 2

computational properties of three term recurrence relations for kummer functions several three term recurrence relations for confluent hypergeometric functions are analyzed from a numerical point of view minimal and dominant solutions for complex values of the variable z are given derived from asymptotic estimates of the whittaker functions with large parameters the laguerre polynomials and the regular coulomb wave functions are studied as particular cases with numerical examples of their computation

asymptotics of orthogonal polynomials generated by a geronimus perturbation of the laguerre measure abstract this paper deals with monic orthogonal polynomials generated by a geronimus canonical spectral transformation of the laguerre classical measure i e 1 x c x e x d x n x c for x 0 1 a free parameter n r and a shift c 0 we analyze the asymptotic behavior both strong and relative to classical laguerre polynomials of these orthogonal polynomials as n tends to infinity

a characteristic galerkin approximation to a system of shallow water equations summary characteristic methods are known to handle advective flow better than traditional galerkin methods and allow large time steps to be taken when compared to standard time stepping methods in this paper we investigate a characteristic galerkin approximation to the 2 dimensional system of shallow water equations we derive cal l infty left 0 t cal l 2 omega right bounds for elevation and velocity showing these to be optimal for velocity in cal l 2 left 0 t cal h 1 omega right

the exponentially convergent trapezoidal rule it is well known that the trapezoidal rule converges geometrically when applied to analytic functions on periodic intervals or the real line the mathematics and history of this phenomenon are reviewed and it is shown that far from being a curiosity it is linked with computational methods all across scientific computing including algorithms related to inverse laplace transforms special functions complex analysis rational approximation integral equations and the computation of functions and eigenvalues of matrices and operators

fast and accurate computation of gauss legendre and gauss jacobi quadrature nodes and weights an efficient algorithm for the accurate computation of gauss legendre and gauss jacobi quadrature nodes and weights is presented the algorithm is based on newton s root finding method with initial guesses and function evaluations computed via asymptotic formulae the n point quadrature rule is computed in mathcal o n operations to an accuracy of essentially double precision for any n geq 100

the numerical computation of the confluent hypergeometric functionu a b z an algorithm is given for the computation of the confluent hypergeometric functionu a b z for real values ofa b andz z 0 algol 60 procedures are given the computations are based on a miller algorithm and on asymptotic expansions

the abc of hyper recursions each member of the family of gauss hypergeometric functions fn 2f1 a e1n b e2n c e3n z where a b c and z do not depend on n and ej 0 1 not all ej equal to zero satisfies a second order linear difference equation of the form anfn 1 bnfn cnfn 1 0 because of symmetry relations and functional relations for the gauss functions the set of 26 cases for different ej values can be reduced to a set of 5 basic forms of difference equations in this paper the coefficients an bn and cn of these basic forms are given in addition domains in the complex z plane are given where a pair of minimal and dominant solutions of the difference equation have to be identified the determination of such a pair asks for a detailed study of the asymptotic properties of the gauss functions fn for large values of n and of other gauss functions outside this group this will be done in a later paper

sums of automorphisms of free modules and completely decomposable groups abstract for certain groups and modules we discuss the property that every endomorphism of that group or module is a sum of two automorphisms firstly we consider the general case of a free r module m of countably infinite rank where r is an associative unital ring such that for some positive integer m every free r module of finite rank m has the above property we prove that m also has the property and extend this result to free modules of uncountable rank we deduce when r is an elementary divisor ring that every endomorphism of a free r module of rank greater than 1 is a sum of two automorphisms we apply these results to completely decomposable groups and make some interesting deductions about the expression of each endomorphism of such a group as a sum of a fixed number of automorphisms

on the computation and inversion of the cumulative noncentral beta distribution function the computation and inversion of the noncentral beta distribution bp q x y or the noncentral f distribution a particular case of bp q x y play an important role in different applications in this paper we study the stability of recursions satisfied by bp q x y and its complementary function and describe asymptotic expansions useful for computing the function when the parameters are large we also consider the inversion problem of finding x or y when a value of bp q x y is given we provide approximations to x and y which can be used as starting values of methods for solving nonlinear equations such as newton if higher accuracy is needed

finite element approximations to the system of shallow water equations i continuous time a priori error estimates various sophisticated finite element models for surface water flow based on the shallow water equations exist in the literature gray kolar luettich lynch and westerink have developed a hydrodynamic model based on the generalized wave continuity equation gwce formulation and have formulated a galerkin finite element procedure based on combining the gwce with the nonconservative momentum equations numerical experiments suggest that this method is robust and accurate and suppresses spurious oscillations which plague other models we analyze a slightly modified galerkin model which uses the conservative momentum equations cme for this gwce cme system of equations we present a continuous time a priori error estimate based on an asl 2 projection

on the numerical calculation of the roots of special functions satisfying second order ordinary differential equations we describe a method for calculating the roots of special functions satisfying second order ordinary differential equations it exploits the recent observation that the solutions of equations of this type can be represented via nonoscillatory phase functions even in the high frequency regime our approach requires mathcal o 1 operations per root and achieves near machine precision accuracy moreover despite its great generality our approach is competitive with and in many cases faster than specialized state of the art methods for the construction of gaussian quadrature rules of large orders when it is used in such a capacity the performance of the scheme is illustrated with several numerical experiments

every 2n by 2n complex matrix is a sum of three symplectic matrices abstract let j 2 n 0 i n i n 0 an a m 2 n c is called symplectic if a t j 2 n a j 2 n if n 1 then we show that every matrix in m 2 n c is a sum of two symplectic matrices if n 1 then we show that every matrix in m 2 n c is a sum of three symplectic matrices moreover we show that some matrices cannot be written with less than three symplectic matrices we also show that for every a m 2 n c there exist symplectic p q m 2 n c and b c d m n c such that p a q b c 0 d if a is skew hamiltonian j 2 n 1 a t j 2 n a then we show that a is a sum of two symplectic matrices

a multiscale discontinuous galerkin method with the computational structure of a continuous galerkin method proliferation of degrees of freedom has plagued discontinuous galerkin methodology from its inception over 30 years ago this paper develops a new computational formulation that combines the advantages of discontinuous galerkin methods with the data structure of their continuous galerkin counterparts the new method uses local element wise problems to project a continuous finite element space into a given discontinuous space and then applies a discontinuous galerkin formulation the projection leads to parameterization of the discontinuous degrees of freedom by their continuous counterparts and has a variational multiscale interpretation this significantly reduces the computational burden and at the same time little or no degradation of the solution occurs in fact the new method produces improved solutions compared with the traditional discontinuous galerkin method in some situations

fast reliable and unrestricted iterative computation of gauss hermite and gauss laguerre quadratures methods for the computation of classical gaussian quadrature rules are described which are effective both for small and large degree these methods are reliable because the iterative computation of the nodes has guaranteed convergence and they are fast due to their fourth order convergence and its asymptotic exactness for an appropriate selection of the variables for gauss hermite and gauss laguerre quadratures local taylor series can be used for computing efficiently the orthogonal polynomials involved with exact initial values for the hermite case and first values computed with a continued fraction for the laguerre case the resulting algorithms have almost unrestricted validity with respect to the parameters full relative precision is reached for the hermite nodes without any accuracy loss and for any degree and a mild accuracy loss occurs for the hermite and laguerre weights as well as for the laguerre nodes these fast methods are exclusively based on convergent processes which together with the high order of convergence of the underlying iterative method makes them particularly useful for high accuracy computations we show examples of very high accuracy computations of up to 1000 digits of accuracy

unconditional and optimal h1 error estimate of a crank nicolson finite difference scheme for the gross pitaevskii equation with an angular momentum rotation term abstract this paper is concerned with the time step condition or grid ratio condition of finite difference method for solving the gross pitaevskii equation with an angular momentum rotation term an optimal error estimate of the finite difference method in h 1 norm is established without any constraints of the grid ratios besides the standard energy method a cut off function technique and a lifting technique is introduced in analyzing the proposed scheme the analysis method used in this paper can be applied to many other finite difference schemes for solving the nonlinear schrodinger type equations for which previous works often require certain restriction on the grid ratios numerical results are reported to verify the error estimates and conservation laws

iteration free computation of gauss legendre quadrature nodes and weights gauss legendre quadrature rules are of considerable theoretical and practical inter est because of their role in numerical integration and interpolation in this paper a series expansion for the zeros of the legendre polynomials is constructed in addition a series expansion useful for the computation of the gauss legendre weights is derived together these two expansions provide a practical and fast iteration free method to compute individual gauss legendre node weight pairs in o 1 complexity and with double precision accuracy an expansion for the barycentric interpolation weights for the gauss legendre nodes is also derived a c implementation is available online

finite element approximations to the system of shallow water equations part iii on the treatment of boundary conditions we continue our investigation of finite element approximations to the system of shallow water equations based on the generalized wave continuity equation gwce formulation in previous work we analyzed this system assuming dirichlet boundary conditions on both elevation and velocity based on physical grounds it is possible to not impose boundary conditions on elevation thus we examine the formulation for the case of dirichlet conditions on velocity only the changes required to the finite element method are presented and stability and error estimates are derived for an approximate linear model assuming continuous time stability for a discrete time method is also shown

a galerkin method with spherical splines for the shallow water equations on a sphere error analysis we analyse a galerkin method for the viscous shallow water equations on the unit sphere using spherical splines a priori error estimates for semi discrete and fully discrete schemes are derived in particular the proposed fully discrete scheme has second order accuracy in time numerical experiments support our theoretical results

discontinuous and coupled continuous discontinuous galerkin methods for the shallow water equations we consider the approximation of a simplified model of the depth averaged two dimensional shallow water equations by two approaches in both approaches a discontinuous galerkin dg method is used to approximate the continuity equation in the first approach a continuous galerkin method is used for the momentum equations in the second approach a particular dg method the nonsymmetric interior penalty galerkin method is used to approximate momentum a priori error estimates are derived and numerical results are presented for both approaches

numerically satisfactory solutions of kummer recurrence relations pairs of numerically satisfactory solutions as n rightarrow infty for the three term recurrence relations satisfied by the families of functions 1 rm f 1 a epsilon 1 n b epsilon 2 n z epsilon i in mathbb z are given it is proved that minimal solutions always exist except when epsilon 2 0 and z is in the positive or negative real axis and that 1 rm f 1 a epsilon 1 n b epsilon 2 n z is minimal as n rightarrow infty whenever epsilon 2 0 the minimal solution is identified for any recurrence direction that is for any integer values of epsilon 1 and epsilon 2 when epsilon 2 neq 0 the confluent limit lim b rightarrow infty 1 rm f 1 gamma b b z e gamma z with gamma in mathbb c fixed is the main tool for identifying minimal solutions together with a connection formula for epsilon 2 0 lim a rightarrow infty 1 rm f 1 a b z 0 rm f 1 b az e z 2 is the main tool to be considered

a discontinuous hp finite element method for convection diffusion problems this paper presents a new method which exhibits the best features of both finite volume and finite element techniques special attention is given to the issues of conservation flexible accuracy and stability the method is elementwise conservative the order of polynomial approximation can be adjusted element by element and the stability is not based on the introduction of artificial diffusion but on the use of a very particular finite element formulation with discontinuous basis functions the method supports h p and hp approximations and can be applied to any type of meshes including non matching grids a priori error estimates and numerical experiments on representative model problems indicate that the method is robust and capable of delivering high accuracy

an explicit hybridized discontinuous galerkin method for serre green naghdi wave model abstract simulation of coastal water waves requires a relevant mathematical model and a proper numerical technique which can be used in irregular geometries in this paper we study the numerical solution to a class of irrotational green naghdi gn equation using the hybridized discontinuous galerkin method hdg we first use strang splitting to decompose this problem into a hyperbolic and a dispersive sub problem for the hyperbolic sub problem we use an explicit hdg with lax friedrichs numerical fluxes we also include different types of boundary conditions in our formulation similar to other hdg implementations our global unknowns are the numerical traces i e the unknowns on the mesh skeleton this choice can significantly reduce the number of global degrees of freedom solving this part of the problem involves newton iterations at each time step however unlike the implicit versions of hdg the element equations are decoupled and we do not need to solve a large global system of equations on the other hand for the dispersive sub problem we need to solve a global system of equations implicitly since hdg only involves unknowns on the mesh skeleton it can be a very efficient alternative for this implicit solve step by coupling these two solvers we obtain the solution to the green naghdi equation next we use this technique to solve a set of verification and validation examples in the first example we show the convergence properties of the numerical method next we compare our results with a set of experimental data for the nonlinear water waves in different situations finally we present a two dimensional simulation result for a solitary wave passing over a cosine bump we observe close to optimal convergence rates and a good agreement between our numerical results and the experimental data

asymptotic inversion of the erlang b formula the erlang b formula represents the steady state blocking probability in the erlang loss model or m m s s queue we derive asymptotic expansions for the offered load that matches for a given number of servers a certain blocking probability in addressing this inversion problem we make use of various asymptotic expansions for the incomplete gamma function a similar inversion problem is investigated for the erlang c formula

accurate computation of weights in classical gauss christoffel quadrature rules for many classical gauss christoffel quadrature rules there does not exist a method which guarantees a uniform level of accuracy for the gaussian quadrature weights at all quadrature nodes unless the nodes are known exactly more disturbing some algebraic expressions for these weights exhibit an excessive sensitivity to even the smallest perturbations in the node location this sensitivity rapidly increases with high order quadrature rules current uses of very high order quadratures are common with the advent of more powerful computers and a loss of accuracy in the weights has become a problem and must be addressed a simple but efficient and general method for improving the accuracy of the computation of the quadrature weights is proposed it ensures a high level of accuracy for these weights even though the nodes may carry a significant large error in addition a highly efficient root finding iterative technique with superlinear converging rates for computing the nodes is developed it uses solely the quadrature polynomials and their first derivatives a comparison of this method with the eigenvalue method of golub and welsh implemented in most standard software libraries is made the proposed method outperforms the latter from the point of view of both accuracy and efficiency the legendre lobatto radau hermite and laguerre quadrature rules are examined

finite element approximations of the nonstationary navier stokes problem part iv error estimates for second order time discretization this paper provides an error analysis for the crank nicolson method of time discretization applied to spatially discrete galerkin approximations of the nonstationary navier stokes equations second order error estimates are proven locally in time under realistic assumptions about the regularity of the solution for approximations of an exponentially stable solution these local error estimates are extended uniformly in time as text t to infty

on computing the points and weights for gauss legendre quadrature three methods are compared for computing n points xi and weights w xi for gauss legendre quadrature the eigensystem method computes the points as the eigenvalues of a symmetric tridiagonal matrix and the weights are computed from its eigenvectors the resulting error is o n for the points and o n 2 for the weights using an inverse taylor series computation for the points yakimiw j comput phys 129 1996 pp 406 430 reduced the point error growth to o 1 he also reduced the error in the weights to o n and obtained uniform pointwise relative accuracy however the points xi cluster quadratically near the end points x pm 1 where for some n nmax they coalesce or even exchange to within machine accuracy and computation fails although this has not been a problem in the past it could soon become so because weather models are now approaching resolutions with n 1000 here we compare these methods to the fourier newton method which consists of newton s method for computing the transformed points theta i arccos x i as the zeros of the fourier representations of the legendre polynomials p n theta the theta i are near equally spaced particularly in the neighborhood of x 0 for a fixed arithmetic precision this extends the range of computable points and weights to n n max 2 where nmax is the limit for methods based on xi both the accuracy and speed of the three methods are tabulated

coupled discontinuous and continuous galerkin finite element methods for the depth integrated shallow water equations abstract in this paper we investigate a new approach for the two dimensional fully non linear depth averaged shallow water equations by coupling continuous and discontinuous galerkin methods we apply a discontinuous galerkin method to the primitive continuity and momentum equations and couple this to a continuous method discretizing the so called wave continuity and momentum equation herein we present the formulation and derive an a priori error estimate and demonstrate some preliminary numerical results this work represents the culmination of a series of papers exploring coupled methods for the shallow water equations

computing the kummer function u a b z for small values of the arguments we describe methods for computing the kummer function u a b z for small values of z with special attention to small values of b for these values of b the connection formula that represents u a b z as a linear combination of two 1f1 functions needs a limiting procedure we use the power series of the 1f1 functions and consider the terms for which this limiting procedure is needed we give recursion relations for higher terms in the expansion and we consider the derivative u a b z as well we also discuss the performance for small z of an asymptotic approximation of the kummer function in terms of modified bessel functions

when is every linear transformation a sum of an idempotent one and a locally nilpotent one abstract for a semisimple module m over a ring r with r j r boolean every endomorphism of m is a sum of an idempotent endomorphism and a locally nilpotent endomorphism as a consequence it is proved that for a vector space v over a division ring d every linear transformation of v is a sum of an idempotent linear transformation and a locally nilpotent linear transformation if and only if d f 2 this can be seen as an answer to the local version of a question raised by breaz et al in 1 on nil cleanness of the ring of linear transformations of an infinite dimensional vector space

an a priori error analysis of the local discontinuous galerkin method for elliptic problems in this paper we present the first a priori error analysis for the local discontinuous galerkin ldg method for a model elliptic problem for arbitrary meshes with hanging nodes and elements of various shapes we show that for stabilization parameters of order one the l2 norm of the gradient and the l2 norm of the potential are of order k and k 1 2 respectively when polynomials of total degree at least k are used if stabilization parameters of order h 1 are taken the order of convergence of the potential increases to k 1 the optimality of these theoretical results is tested in a series of numerical experiments on two dimensional domains

computing the confluent hypergeometric function m a b x the confluent hypergeometric function m a b x arises naturally in both statistics and physics although analytically well behaved extreme but practically useful combinations of parameters create extreme computational difficulties a brief review of known analytic and computational results highlights some difficult regions including b a 0 with x much larger than b existing power series and integral representations may fail to converge numerically while asymptotic series representations may diverge before achieving the accuracy desired continued fraction representations help somewhat variable precision can circumvent the problem but with reductions in speed and convenience in some cases known analytic properties allow transforming a difficult computation into an easier one the combination of existing computational forms and transformations still leaves gaps for b a 0 two new power series in terms of gamma and beta cumulative distribution functions respectively help in some cases numerical evaluations highlight the abilities and limitations of existing and new methods overall a rational approximation due to luke and the new gamma based series provide the best performance

compact finite difference schemes with high accuracy for one dimensional nonlinear schrodinger equation abstract in this paper two compact finite difference schemes are presented for the numerical solution of the one dimensional nonlinear schrodinger equation the discrete l 2 norm error estimates show that convergence rates of the present schemes are of order o h 4 2 numerical experiments on some model problems show that the present schemes preserve the conservation laws of charge and energy and are of high accuracy

every endomorphism of a local warfield module of finite torsion free rank is the sum of two automorphisms abstract a well known result of p hill s 1969 trans amer math soc 141 99 105 says that any endomorphism of a totally projective abelian p group for any odd prime is the sum of two automorphisms we will extend this result to local warfield modules of finite torsion free rank over odd primes

o 1 computation of legendre polynomials and gauss legendre nodes and weights for parallel computing a self contained set of algorithms is proposed for the fast evaluation of legendre polynomials of arbitrary degree and argument in 1 1 more specifically the time required to evaluate any legendre polynomial regardless of argument and degree is bounded by a constant i e the complexity is mathcal o 1 the proposed algorithm also immediately yields an mathcal o 1 algorithm for computing an arbitrary gauss legendre quadrature node such a capability is crucial for efficiently performing certain parallel computations with high order legendre polynomials such as computing an integral in parallel by means of gauss legendre quadrature and the parallel evaluation of legendre series in order to achieve the mathcal o 1 complexity novel efficient asymptotic expansions are derived and used alongside known results a c implementation is available from the authors that includes the evaluation routines of the legendre polynomials and gauss legendre quadrature rules

a discontinuoushpfinite element method for diffusion problems we present an extension of the discontinuous galerkin method which is applicable to the numerical solution of diffusion problems the method involves a weak imposition of continuity conditions on the solution values and on fluxes across interelement boundaries within each element arbitrary spectral approximations can be constructed with different orderspin each element we demonstrate that the method is elementwise conservative a property uncharacteristic of high order finite elements for clarity we focus on a model class of linear second order boundary value problems and we developpriorierror estimates convergence proofs and stability estimates the results of numerical experiments onh andp convergence rates for representative two dimensional problems suggest that the method is robust and capable of delivering exponential rates of convergence

a stabilized mixed discontinuous galerkin method for darcy flow abstract a new mixed stabilized discontinuous galerkin formulation for darcy flow is presented the formulation combines several attributes not simultaneously satisfied by other methods it is convergent for any combination of velocity and pressure interpolation higher than first order it exactly satisfies a mass balance on each element and it passes two and three dimensional constant flow patch tests for distorted element geometries the key ingredient in the formulation is a volumetric residual based stabilization term that does not involve any mesh dependent parameters

discontinuous continuous galerkin methods for coupling the primitive and wave continuity equations of shallow water in this paper we investigate a new approach for the numerical solution of the two dimensional depth integrated shallow water equations based on coupling discontinuous and continuous galerkin methods in this approach we couple a discontinuous galerkin method applied to the primitive continuity equation coupled to a continuous galerkin method applied to the so called wave continuity equation the formulation is derived and a priori error estimates are given preliminary numerical results for a simple model of flow in a channel are presented 2003 elsevier b v all rights reserved

an analysis of the wave equation model for finite element tidal computations abstract the one dimensional linearized version of the wave equation model developed by gray and lynch for solving the shallow water equations is analysed similarities with a mixed interpolation finite element method are discovered and the proposed time stepping methods are shown to be a subset of a much larger class using dispersion and asymptotic analyses particular time stepping methods which most accurately represent wave propagation and wave amplitude growth are determined for both the lumped and unlumped approaches

nil clean rings abstract many variations of the notions of clean and strongly clean have been studied by a variety of authors we develop a general theory based on idempotents and direct sum decompositions that unifies several of these existing concepts as a specific case we also investigate a new class of clean rings

transitory minimal solutions of hypergeometric recursions and pseudoconvergence of associated continued fractions three term recurrence relations yn 1 bnyn anyn 1 0 can be used for computing recursively a great number of special functions depending on the asymptotic nature of the function to be computed different recursion directions need to be considered backward for minimal solutions and forward for dominant solutions however some solutions interchange their role for finite values of n with respect to their asymptotic behaviour and certain dominant solutions may transitorily behave as minimal this phenomenon related to gautschi s anomalous convergence of the continued fraction for ratios of confluent hypergeometric functions is shown to be a general situation which takes place for recurrences with an negative and bn changing sign once we analyze the anomalous convergence of the associated continued fractions for a number of different recurrence relations modified bessel functions confluent and gauss hypergeometric functions and discuss the implication of such transitory behaviour on the numerical stability of recursion

coupling of continuous and discontinuous galerkin methods for transport problems in this paper we formulate a coupled discontinuous continuous galerkin method for the numerical solution of convection diffusion transport equations where convection may be dominant one motivation for this approach is to use a discontinuous method where the solution is rough e g in regions of high gradients and use a continuous method where the solution is smooth in this approach the domain is decomposed into two regions and appropriate transmission conditions are applied at the interface between regions in one region a local discontinuous galerkin method is applied and in the other region a standard continuous galerkin method is employed stability and a priori error estimates for the coupled method are derived and numerical results in one space dimension are given for smooth problems and problems with sharp fronts 2002 elsevier science b v all rights reserved

identifying minimal and dominant solutions for kummer recursions we identify minimal and dominant solutions of three term recurrence relations for the confluent hypergeometric functions 1f1 a 1n c 2n z and u a 1n c 2n z where i 0 1 not both equal to 0 the results are obtained by applying perron s theorem together with uniform asymptotic estimates derived by t m dunster for whittaker functions with large parameter values the approximations are valid for complex values of a c and z with ph z

hp version discontinuous galerkin methods for hyperbolic conservation laws abstract the development of hp version discontinuous galerkin methods for hyperbolic conservation laws is presented in this work a priori error estimates are derived for a model class of linear hyperbolic conservation laws these estimates are obtained using a new mesh dependent norm that reflects the dependence of the approximate solution on the local element size and the local order of approximation the results generalize and extend previous results on mesh dependent norms to hp version discontinuous galerkin methods a posteriori error estimates which provide bounds on the actual error are also developed in this work numerical experiments verify the a priori estimates and demonstrate the effectiveness of the a posteriori estimates in providing reliable estimates of the actual error in the numerical solution