pores resolving simulation of darcy flows a theoretical formulation and corresponding numerical solutions are presented for microscopic fluid flows in porous media with the domain sufficiently large to reproduce integral darcy scale effects pore space geometry and topology influence flow through media but the difficulty of observing the configurations of real pore spaces limits understanding of their effects a rigorous direct numerical simulation dns of percolating flows is a formidable task due to intricacies of internal boundaries of the pore space representing the grain size distribution by means of repelling body forces in the equations governing fluid motion greatly simplifies computational efforts an accurate representation of pore scale geometry requires that within the solid the repelling forces attenuate flow to stagnation in a short time compared to the characteristic time scale of the pore scale flow in the computational model this is achieved by adopting an implicit immersed boundary method with the attenuation time scale smaller than the time step of an explicit fluid model a series of numerical simulations of the flow through randomly generated media of different porosities show that computational experiments can be equivalent to physical experiments with the added advantage of nearly complete observability besides obtaining macroscopic measures of permeability and tortuosity numerical experiments can shed light on the effect of the pore space structure on bulk properties of darcy scale flows

a multipoint flux mixed finite element method we develop a mixed finite element method for single phase flow in porous media that reduces to cell centered finite differences on quadrilateral and simplicial grids and performs well for discontinuous full tensor coefficients motivated by the multipoint flux approximation method where subedge fluxes are introduced we consider the lowest order brezzi douglas marini bdm mixed finite element method a special quadrature rule is employed that allows for local velocity elimination and leads to a symmetric and positive definite cell centered system for the pressures theoretical and numerical results indicate second order convergence for pressures at the cell centers and first order convergence for subedge fluxes second order convergence for edge fluxes is also observed computationally if the grids are sufficiently regular

generalized apollonian packings in this paper we generalize the classical two dimensional apollonian packing of circles to the case where the circles are no more tangent we introduce two elements ofsl 2 as generators r andt that are hyperbolic rotations of 2 3 and 2 n n 2 3 4 around two distinct points the limit set of the discrete group generated byr andt provides forn 7 8 9 a generalization of the apollonian packing which is itself recovered forn the valuesn 2 3 4 5 produce a very different result giving rise to the rotation groups of the cube forn 2 and 4 and the icosahedron forn 3 and 5 forn 6 the group is no longer discrete to further analyze this structure forn 7 we move to the minkowski space in which the group acts on a one sheeted hyperboloid the circles are now represented by points on this variety and generate a crystal on it

the numerical solution of diffusion problems in strongly heterogeneous non isotropic materials a new second order finite difference algorithm for the numerical solution of diffusion problems in strongly heterogeneous and non isotropic media is constructed on problems with rough coefficients or highly nonuniform grids the new algorithm is superior to all other algorithms we have compared it with for problems with smooth coefficients on smooth grids the method is comparable with other second order methods the new algorithm is formulated for logically rectangular grids and is derived using thesupport operators method a key idea in deriving the method was to replace the usual inner product of vector functions by an inner product weighted by the inverse of the material properties tensor and to use the flux operator defined as the material properties tensor times the gradient rather than the gradient as one of the basic first order operators in the support operators method the discrete analog of the flux operator must also be the negative adjoint of the discrete divergence in an inner product that is a discrete analog of the continuum inner product the resulting method is conservative and the discrete analog of the variable coefficient laplacian is symmetric and negative definite on nonuniform grids in addition on any grid the discrete divergence is zero on constant vectors the null space for the gradient is the constant functions and when the material properties are piecewise constant the discrete flux operator is exact for piecewise linear functions we compare the methods on some of the most difficult examples to be found in the literature

a family of mpfa finite volume schemes with full pressure support for the general tensor pressure equation on cell centered triangular grids a new family of cell centered finite volume schemes is presented for solving the general full tensor pressure equation of subsurface flow in porous media on arbitary unstructured triangulations the new schemes are flux continuous and have full pressure support fps over each subcell with continuous pressure imposed across each control volume sub interface in contrast to earlier formulations the earlier methods are point wise continuous in pressure and flux with triangle pressure support tps which leads to a more limited quadrature range an m matrix analysis identifies bounding limits for the schemes to posses a local discrete maximum principle conditions for the schemes to be positive definite are also derived a range of computational examples are presented for unstructured triangular grids including highly irregular grids and the new fps schemes are compared against the earlier pointwise continuous tps formulations the earlier pointwise tps methods can induce strong spurious oscillations for problems involving strong full tensor anisotropy where the m matrix conditions are violated and can lead to decoupled solutions in such cases unstructured cell centered decoupling is investigated in contrast to tps the new fps formulation leads to well resolved solutions that are essentially free of spurious oscillations a substantial degree of improved convergence behavior for both pressure and velocity is also observed in all convergence tests this is particularly important for problems involving high anisotropy ratios also the new formulation proves to be highly beneficial for an upscaling example where enhancement of convergence is highly significant for certain quadrature points clearly demonstrating further advantages of the new formulation

zero sets of bivariate hermite polynomials abstract we establish various properties for the zero sets of three families of bivariate hermite polynomials special emphasis is given to those bivariate orthogonal polynomials introduced by hermite by means of a rodrigues type formula related to a general positive definite quadratic form for this family we prove that the zero set of the polynomial of total degree n m consists of exactly n m disjoint branches and possesses n m asymptotes a natural extension of the notion of interlacing is introduced and it is proved that the zero sets of the family under discussion obey this property the results show that the properties of the zero sets considered as affine algebraic curves in r 2 are completely different for the three families analyzed

orthogonal polynomial eigenfunctions of second order partial differerential equations in this paper we show that for several second order partial differential equations l u a x y uxx 2b x y uxy c x y uyy d x y ux e x y uy atu which have orthogonal polynomial eigenfunctions these polynomials can be expressed as a product of two classical orthogonal polynomials in one variable this is important since otherwise it is very difficult to explicitly find formulas for these polynomial solutions from this observation and characterization we are able to produce additional examples of such orthogonal polynomials together with their orthogonality that widens the class found by h l krall and sheffer in their seminal work in 1967 moreover from our approach we can answer some open questions raised by krall and sheffer

weak galerkin finite element methods for darcy flow anisotropy and heterogeneity this paper presents a family of weak galerkin finite element methods wgfems for darcy flow computation the wgfems are new numerical methods that rely on the novel concept of discrete weak gradients the wgfems solve for pressure unknowns both in element interiors and on the mesh skeleton the numerical velocity is then obtained from the discrete weak gradient of the numerical pressure the new methods are quite different than many existing numerical methods in that they are locally conservative by design the resulting discrete linear systems are symmetric and positive definite and there is no need for tuning problem dependent penalty factors we test the wgfems on benchmark problems to demonstrate the strong potential of these new methods in handling strong anisotropy and heterogeneity in darcy flow

multifractality of space filling bearings and apollonian packings we study the structure of space filling bearings sfb and apollonian packings ap by determining the scaling behavior of the density distribution of points where the circles touch each other application of the sand box method to the average of theqth power of the number of touching points reveals the multifractal aspects of the structure of sfb and ap

on bivariate classical orthogonal polynomials abstract we deduce new characterizations of bivariate classical orthogonal polynomials associated with a quasi definite moment functional and we revise old properties for these polynomials more precisely new characterizations of classical bivariate orthogonal polynomials satisfying a diagonal pearson type equation are proved they are solutions of two separate partial differential equations one for every partial derivative their partial derivatives are again orthogonal and every vector polynomial can be expressed in terms of its partial derivatives by means of a linear relation involving only three terms of consecutive total degree moreover we study general solutions of the matrix second order partial differential equation satisfied by classical orthogonal polynomials and we deduce the explicit expressions for the matrix coefficients of the structure relation finally some illustrative examples are given

partial differential equations having orthogonal polynomial solutions abstract we show that if a second order partial differential equation l u auxx 2buxy cuyy dux euy nu has orthogonal polynomial solutions then the differential operator l must be symmetrizable and can not be parabolic in any nonempty open subset of the plane we also find rodrigues type formula for orthogonal polynomial solutions of such differential equations

on a class of bivariate second order linear partial difference equations and their monic orthogonal polynomial solutions abstract in this paper we classify the bivariate second order linear partial difference equations which are admissible potentially self adjoint and of hypergeometric type using vector matrix notation explicit expressions for the coefficients of the three term recurrence relations satisfied by monic orthogonal polynomial solutions are obtained in terms of the coefficients of the partial difference equation finally we make a compilation of the examples existing in the literature belonging to the class analyzed in this paper namely bivariate charlier meixner kravchuk and hahn orthogonal polynomials

m matrix flux splitting for general full tensor discretization operators on structured and unstructured grids diagonal tensor flux approximations are commonly used in fluid dynamics this approximation introduces an o 1 error in flux whenever the coordinate system is nonaligned with the principal axes of the tensor which is particularly common when employing curvilinear gridding in general a consistent full tensor flux approximation leads to a significant increase in support and consequent size of the jacobian matrix after decomposition of a general full tensor flux into a diagonal tensor flux together with cross terms time split semi implicit stable full tensor flux approximations are introduced with in a general finite volume formalism enabling the standard diagonal tensor jacobian matrix structure to be retained for single phase flow impes and standard block fully implicit formulations while ensuring spatial consistency of the discretization for both structured and unstructured grids stability of the scheme is proven for constant elliptic coefficients the results presented demonstrate the benefits of the method for multiphase flow within a fully implicit framework on structured and unstructured grids

structure relations for the bivariate big q jacobi polynomials we give structure relations for the orthogonal polynomials in two variables defined by lewanowicz and wozny s lewanowicz p wozny j comput appl math 233 2010 1554 1561 p n k x y a b c d q p n k y a bcq 2 k 1 dq k q y kdq y q kp kx y c b d y q n 0 k 0 1 n where email protected 0 1 0

general recurrence and ladder relations of hypergeometric type functions a method for the explicit construction of general linear sum rules involving hypergeometric type functions and their derivatives of any order is developed this method only requires the knowledge of the coefficients of the differential equation that they satisfy namely the hypergeometric type differential equation special attention is paid to the differential recurrence or ladder relations and to the fundamental three term recurrence formulas most recurrence and ladder relations published in the literature for numerous special functions including the classical orthogonal polynomials are instances of these sum rules moreover an extension of the method to the generalized hypergeometric type functions is also described allowing us to obtain explicit ladder operators for the radial wave functions of multidimensional hydrogen like atoms where the varying parameter is the dimensionality

mixed finite elements for elliptic problems with tensor coefficients as cell centered finite differences we present an expanded mixed finite element approximation of second order elliptic problems containing a tensor coefficient the mixed method is expanded in the sense that three variables are explicitly approximated namely the scalar unknown the negative of its gradient and its flux the tensor coefficient times the negative gradient the resulting linear system is a saddle point problem in the case of the lowest order raviart thomas elements on rectangular parallelepipeds we approximate this expanded mixed method by incorporating certain quadrature rules this enables us to write the system as a simple cell centered finite difference method requiring the solution of a sparse positive semidefinite linear system for the scalar unknown for a general tensor coefficient the sparsity pattern for the scalar unknown is a 9 point stencil in two dimensions and 19 points in three dimensions existing theory shows that the expanded mixed method gives optimal order approximations in the l 2 and h s norms and superconvergence is obtained between the l 2 projection of the scalar variable and its approximation we show that these rates of convergence are retained for the finite difference method if h denotes the maximal mesh spacing then the optimal rate is o h the superconvergence rate o h 2 is obtained for the scalar unknown and rate o h 3 2 for its gradient and flux in certain discrete norms moreover the full o h 2 is obtained in the strict interior of the domain computational results illustrate these theoretical results

apollonian arrangements of spheres in d dimensional space the problem of arranging d dimensional spheres in such a way that each sphere touches exactly d 1 others is formulated by means of linear recursions involving d 1 different d 2 d 2 matrices the average curvatures are calculated exactly for any dimension special scaling limits for the powers of one matrix or of the products of two matrices show interesting behaviour the corresponding curvatures either diverge quadratically or form a six step cycle only for d 3 and one matrix or show ergodic behaviour sometimes with universal invariant measure the latter type of behaviour is also found if the results are formally extended to non integer values of d

linear partial q difference equations on q linear lattices and their bivariate q orthogonal polynomial solutions orthogonal polynomial solutions of an admissible potentially self adjoint linear second order partial q difference equation of the hypergeometric type in two variables on q linear lattices are analyzed a q pearson s system for the orthogonality weight function as well as for the difference derivatives of the solutions are presented giving rise to a solution of the q difference equation under study in terms of a rodrigues type formula the monic orthogonal polynomial solutions are treated in detail giving explicit formulae for the matrices in the corresponding recurrence relations they satisfy lewanowicz and wozny s lewanowicz p wozny j comput appl math 233 2010 1554 1561 have recently introduced a non monic bivariate extension of big q jacobi polynomials together with a partial q difference equation of the hypergeometric type that governs them this equation is analyzed in the last section we provide two more orthogonal polynomial solutions namely a second non monic solution from the rodrigues representation and the monic solution both from the recurrence relation that govern them and also explicitly given in terms of generalized bivariate basic hypergeometric series limit relations as email protected for the partial q difference equation and for the all three q orthogonal polynomial solutions are also presented

monotone finite volume schemes for diffusion equations on unstructured triangular and shape regular polygonal meshes we consider a non linear finite volume fv scheme for stationary diffusion equation we prove that the scheme is monotone i e it preserves positivity of analytical solutions on arbitrary triangular meshes for strongly anisotropic and heterogeneous full tensor coefficients the scheme is extended to regular star shaped polygonal meshes and isotropic heterogeneous coefficients

the hausdorf dimension of the apollonian packing of circles we formulate the problem of determining the hausdorf dimension df of the apollonian packing of circles as an eigenvalue problem of a linear integral equation we show that solving a finite dimensional approximation to this infinite order matrix equation and extrapolating the results provides a fast algorithm for obtaining high precision numerical estimates for df we find that df 1 305 686 729 10 this is consistent with the rigorously known bounds on df and improves the precision of the existing estimate by three orders of magnitude

a multipoint flux mixed finite element method on distorted quadrilaterals and hexahedra in this paper we develop a new mixed finite element method for elliptic problems on general quadrilateral and hexahedral grids that reduces to a cell centered finite difference scheme a special non symmetric quadrature rule is employed that yields a positive definite cell centered system for the pressure by eliminating local velocities the method is shown to be accurate on highly distorted rough quadrilateral and hexahedral grids including hexahedra with non planar faces theoretical and numerical results indicate first order convergence for the pressure and face fluxes

a quasi positive family of continuous darcy flux finite volume schemes with full pressure support a new family of flux continuous locally conservative finite volume schemes is presented for solving the general tensor pressure equation of subsurface flow in porous media the new schemes have full pressure continuity imposed across control volume faces previous families of flux continuous schemes are point wise continuous in pressure and flux when applying the earlier point wise flux continuous schemes to strongly anisotropic full tensor fields their failure to satisfy a maximum principle as with other fem and finite volume methods can result in loss of local stability for high anisotropy ratios which can cause strong spurious oscillations in the numerical pressure solution an m matrix analysis reveals the upper limits for guaranteeing a maximum principle for general 9 point schemes and aids in the design of schemes that minimize the occurrence of spurious oscillations in the discrete pressure field the full pressure continuity schemes are shown to possess a larger range of flux continuous schemes than the previous point wise counter parts for strongly anisotropic full tensor cases it is shown that the full quadrature range possessed by the new schemes permits these schemes to exploit quadrature points previously out of range that are shown to minimize spurious oscillations in discrete pressure solutions the new formulation leads to a more robust quasi positive family of flux continuous schemes applicable to general discontinuous full tensor fields

monotonicity of control volume methods robustness of numerical methods for multiphase flow problems in porous media is important for development of methods to be used in a wide range of applications here we discuss monotonicity for a simplified problem of single phase flow but where the simulation grids and media are allowed to be general posing challenges to control volume methods we discuss discrete formulations of the maximum principle and derive sufficient criteria for discrete monotonicity for arbitrary nine point control volume discretizations for conforming quadrilateral grids in 2d these criteria are less restrictive than the m matrix property it is shown that it is impossible to construct nine point methods which unconditionally satisfy the monotonicity criteria when the discretization satisfies local conservation and exact reproduction of linear potential fields numerical examples are presented which show the validity of the criteria for monotonicity further the impact of nonmonotonicity is studied different behavior for different discretization methods is illuminated and simple ideas are presented for improvement in terms of monotonicity

on limit relations between some families of bivariate hypergeometric orthogonal polynomials in this paper we deal with limit relations between bivariate hypergeometric polynomials we analyze the limit relation from trinomial distribution to bivariate gaussian distribution obtaining the limit transition from the second order partial difference equation satisfied by bivariate hypergeometric kravchuk polynomials to the second order partial differential equation verified by bivariate hypergeometric hermite polynomials as a consequence the limit relation between both families of orthogonal polynomials is established a similar analysis between bivariate hahn and bivariate appell orthogonal polynomials is also presented

recursive computation of generalised zernike polynomials abstract an algorithmic approach for generating generalised zernike polynomials by differential operators and connection matrices is proposed this is done by introducing a new order of generalised zernike polynomials such that it collects all the polynomials of the same total degree in a column vector the connection matrices between these column vectors composed by the generalised zernike polynomials and a family of polynomials generated by a rodrigues formula are given explicitly this yields a rodrigues type formula for the generalised zernike polynomials themselves with properly defined differential operators another consequence of our approach is the fact that the generalised zernike polynomials obey a rather simple partial differential equation we recall also how to define hermite zernike polynomials

finite volume scheme for anisotropic diffusion in this paper we apply a special finite volume scheme limited to smooth temperature distributions and cartesian grids to test the importance of connectivity of the finite volumes the area of application is nuclear fusion plasma with field line aligned temperature gradients and extreme anisotropy we apply the scheme to the anisotropic heat conduction equation and compare its results with those of existing finite volume schemes for anisotropic diffusion also we introduce a general model adaptation of the steady diffusion equation for extremely anisotropic diffusion problems with closed field lines

monotonic solution of heterogeneous anisotropic diffusion problems anisotropic problems arise in various areas of science and engineering for example groundwater transport and petroleum reservoir simulations the pure diffusive anisotropic time dependent transport problem is solved on a finite number of nodes that are selected inside and on the boundary of the given domain along with possible internal boundaries connecting some of the nodes an unstructured triangular mesh that attains the generalized anisotropic delaunay condition for all the triangle sides is automatically generated by properly connecting all the nodes starting from an arbitrary initial one the control volume of each node is the closed polygon given by the union of the midpoint of each side with the anisotropic circumcentre of each final triangle a structure of the flux across the control volume sides similar to the standard galerkin finite element scheme is derived a special treatment of the flux computation mainly based on edge swaps of the initial mesh triangles is proposed in order to obtain a stiffness m matrix system that guarantees the monotonicity of the solution the proposed scheme is tested using several literature tests and the results are compared with analytical solutions as well as with the results of other algorithms in terms of convergence order computational costs are also investigated

cross flow tensors and finite volume approximation with by deferred correction abstract a general framework for full tensor finite volume schemes is presented along with properties of the schemes that in turn provide motivation for a deferred correction scheme the approach is illustrated by writing a general nine node discretization as a five node scheme plus a residual term involving the extra nodes a possible solution procedure involves solving the standard five node matrix problem to yield an initial solution then re solving the problem with the residual determined by the first iterate of the pressure field relaxation using the pressure from the previous impes time step is also examined stability of the inner iteration is considered and strategies are discussed this method is coupled with a higher order godunov scheme for integrating the convective components of the flow equations results presented include two phase flow examples with full tensor fields and serve to demonstrate the potential of the method

two variable orthogonal polynomials of big q jacobi type a four parameter family of orthogonal polynomials in two discrete variables is defined for a weight function of basic hypergeometric type the polynomials which are expressed in terms of univariate big q jacobi polynomials form an extension of dunkl s bivariate little q jacobi polynomials c f dunkl orthogonal polynomials in two variables of q hahn and q jacobi type siam j algebr discrete methods 1 1980 137 151 we prove orthogonality property of the new polynomials and show that they satisfy a three term relation in a vector matrix notation as well as a second order partial q difference equation

minimal recurrence relations for connection coefficients between classical orthogonal polynomials discrete case abstract we present a simple approach in order to compute recursively the connection coefficients between two families of classical discrete orthogonal polynomials charlier meixner kravchuk hahn i e the coefficients cm n in the expression p n x n m 0 c m n q m x where pn x and qm x belong to the aforementioned class of polynomials this is scv2 done by adapting a general and systematic algorithm recently developed by the authors to the discrete classical situation moreover extensions of this method allow to give new addition formulae and to estimate cm n asymptotics in limit relations between some families

bivariate orthogonal polynomials in the lyskova class classical orthogonal polynomials in two variables can be characterized as the polynomial solutions of a matrix second order partial differential equation involving matrix polynomial coefficients in this work we study classical orthogonal polynomials in two variables whose partial derivatives satisfy again a second order partial differential equation of the same type

recurrence equations and their classical orthogonal polynomial solutions the classical orthogonal polynomials are given as the polynomial solutions p n x of the differential email protected x y x email protected x y x email protected ny x 0 where s x is a polynomial of at most second degree and t x is a polynomial of first degree in this paper a general method to express the coefficients a n b n and c n of the recurrence equationp n 1 x a nx b n p n x c np n 1 x in terms of the given polynomials s x and t x is used to present an algorithm to determine the classical orthogonal polynomial solutions of any given holonomic three term recurrence equation i e a homogeneous linear three term recurrence equation with polynomial coefficients in a similar way classical discrete orthogonal polynomial solutions of holonomic three term recurrence equations can be determined by considering their corresponding difference email protected x email protected y x email protected x dy x email protected ny x 0 where dy x y x 1 y x and y x y x y x 1 denote the forward and backward difference operators respectively and a similar approach applies to classical q orthogonal polynomials being solutions of the q difference email protected x d qd 1 qy x email protected x d qy x email protected q ny x 0 whered qf x f qx f x q 1 x q 1 denotes the q difference operator

orthogonal polynomials in two variables and second order partial differential equations abstract we study the second order partial differential equations l u au xx 22 bu xy cu yy du x eu y n u which have orthogonal polynomials in two variables as solutions by using formal functional calculus on moment functionals we first give new simpler proofs and improvements of the results by krall and sheffer and littlejohn we then give a two variable version of al salam and chihara s characterization of classical orthogonal polynomials in one variable we also study in detail the case when l belongs to the basic class that is a y c x 0 in particular we characterize all such differential equations which have a product of two classical orthogonal polynomials in one variable as solutions

precise determination of the fractal dimensions of apollonian packing and space filling bearings the authors present high precision numerical values for the fractal dimensions of apollonian packing and various space filling bearings which might have application for mechanical gearworks for turbulence or for tectonic motion they find for the simplest apollonian packing df 1 305 684 or 0 000 010

double families of quasi positive darcy flux approximations with highly anisotropic tensors on structured and unstructured grids this paper focuses on flux continuous pressure equation approximation for strongly anisotropic media previous work on families of flux continuous schemes for solving the general geometry permeability tensor pressure equation has focused on single parameter families these schemes have been shown to remove the o 1 errors introduced by standard two point flux reservoir simulation schemes when applied to full tensor flow approximation improved convergence of the schemes has also been established for specific quadrature points however these schemes have conditional m matrices depending on the strength of the off diagonal tensor coefficients when applied to cases involving full tensors arising from strongly anisotropic media the point wise continuous schemes can fail to satisfy the maximum principle and induce severe spurious oscillations in the numerical pressure solution new double family flux continuous locally conservative schemes are presented for the general geometry permeability tensor pressure equation the new double family formulation is shown to expand on the current single parameter range of existing conditional m matrix schemes the conditional m matrix bounds on a double family formulation are identified for both quadrilateral and triangle cell grids a quasi positive qm matrix analysis is presented that classifies the behaviour of the new schemes with respect to double family quadrature in regions beyond the m matrix bounds the extension to double family quadrature is shown to be beneficial resulting in novel optimal anisotropic quadrature schemes the new methods are applied to strongly anisotropic full tensor field problems and yield results with sharp resolution with only minor or practically zero spurious oscillations

weak classical orthogonal polynomials in two variables orthogonal polynomials in two variables constitute a very old subject in approximation theory usually they are studied as solutions of second order partial differential equations in this work we study two variable orthogonal polynomials associated with a moment functional satisfying the two variable analogue of the pearson differential equation from this approach we derive the extension of some of the usual characterizations of the classical orthogonal polynomials in one variable

bivariate second order linear partial differential equations and orthogonal polynomial solutions in this paper we construct the main algebraic and differential properties and the weight functions of orthogonal polynomial solutions of bivariate second order linear partial differential equations which are admissible potentially self adjoint and of hypergeometric type general formulae for all these properties are obtained explicitly in terms of the polynomial coefficients of the partial differential equation using vector matrix notation moreover rodrigues representations for the polynomial eigensolutions and for their partial derivatives of any order are given as illustration these results are applied to a two parameter monic appell polynomials finally the non monic case is briefly discussed 2011 elsevier inc

a distributional study of discrete classical orthogonal polynomials abstract for the sequences of discrete classical orthogonal polynomials charlier meixner hahn we can find a functional u which satisfies the difference distributional equation u u where and are polynomials of degrees 2 and 1 respectively from this it follows that these polynomials are solutions of a second order difference equation also they can be represented by a rodrigues type formula the sequence of difference polynomials derived from them constitutes an orthogonal polynomial sequence their weight functions satisfy a pearson type difference equation a structure relation pn 1 anpn 2 bnpn 1 cnpn also holds

a semiclassical perspective on multivariate orthogonal polynomials differential properties for orthogonal polynomials in several variables are studied we consider multivariate orthogonal polynomials whose gradients satisfy some quasi orthogonality conditions we obtain several characterizations for these polynomials including the analogues of the semiclassical pearson differential equation the structure relation and a differential difference equation

on the q polynomials a distributed study in this paper we present a uni1ed distributional study of the classical discrete q polynomials in the hahn s sense from the distributional q pearson equation we will deduce many of their properties such as the three term recurrence relations structure relations etc also several characterizations of such q polynomials are presented c

the structure relation for askey wilson polynomials an explicit structure relation for askey wilson polynomials is given this involves a divided q difference operator which is skew symmetric with respect to the askey wilson inner product and which sends polynomials of degree n to polynomials of degree n 1 by specialization of parameters and by taking limits similar structure relations as well as lowering and raising relations can be obtained for other families in the q askey scheme and the askey scheme this is explicitly discussed for jacobi polynomials continuous q jacobi polynomials continuous q ultraspherical polynomials and for big q jacobi polynomials an already known structure relation for this last family can be obtained from the new structure relation by using the three term recurrence relation and the second order q difference formula the results are also put in the framework of a more general theory their relationship with earlier work by zhedanov and bangerezako is discussed there is also a connection with the string equation in discrete matrix models and with the sklyanin algebra

unstructured grid optimization for improved monotonicity of discrete solutions of elliptic equations with highly anisotropic coefficients multipoint flux approximation mpfa techniques are commonly applied for discretizing the porous media flow equations within the context of finite volume numerical procedures although these methods can be applied to heterogeneous anisotropic systems on generally unstructured grids the inverse of the resulting linear operator can suffer from a loss of monotonicity at high permeability anisotropy ratios resulting in spurious oscillations of the pressure solution the purpose of this paper is to develop a method for optimizing unstructured grids in two and three dimensions such that the monotonicity behavior of the mpfa technique is significantly improved the method employs anisotropic triangulation and can be readily combined with permeability upscaling procedures results are presented for a variety of examples and the technique is shown to perform well on problems involving complex grid point distributions and heterogeneous permeability fields with strong anisotropy ratios of o 100 oscillation free pressure solutions are achieved in all cases considered even for examples in which the original grid shows large oscillations

a cell functional minimization scheme for parabolic problem we are interested in a robust and accurate finite volume scheme for 2 d parabolic problems derived from the cell functional minimization approach the scheme has a local stencil is locally conservative treats discontinuity rigorously and leads to a symmetric positive definite linear system since the scheme has both cell centered unknowns and cell edge unknowns the computational cost is an issue and a parallel algorithm is then suggested based on nonoverlapping domain decomposition approach the interface condition is of the dirichlet robin type and has a parameter l by choosing this parameter properly the convergence of the iteration process could be sped up numerical results for linear and nonlinear problems demonstrate the good performance of the cell functional minimization scheme and its parallel version on distorted meshes

the three term recurrence relation and the differentiation formulas for hypergeometric type functions abstract the functions of hypergeometric type are the solutions y y x of the differential equation z y z y y 0 where are polynomials of degrees not higher than 2 and 1 respectively and is a constant here we consider a class of functions of hypergeometric type with the additional condition that 1 2 1 0 being a complex number in general moreover we assume that the coefficients of the polynomials and have no dependence on to this class of functions belong gauss kummer and hermite functions the classical orthogonal polynomials and many other functions encountered in linear and non linear physics we obtain two important structural properties of these functions i the so called three term recurrence relation which correlates three functions of successive orders and ii the differentiation formulas also called ladder or structure relations or even differential recurrence relations which relate the first derivative y z with the functions y z and y 1 z or y 1 z finally these three relationships are applied to the polynomials of hypergeometric type which form a broad subclass of functions y where is a positive integer number and the associated contour is closed for completeness the explicit expressions corresponding to all classical orthogonal polynomials jacobi laguerre hermite and bessel are tabulated

semiclassical orthogonal polynomials in two variables in this work semiclassical orthogonal polynomials in two variables are defined as the orthogonal polynomials associated with a quasi definite linear functional satisfying a matrix pearson type differential equation semiclassical functionals are characterized by means of the analogue of the structure relation in one variable moreover non trivial examples of semiclassical orthogonal polynomials in two variables are given

linear partial difference equations of hypergeometric type orthogonal polynomial solutions in two discrete variables in this paper a systematic study of the orthogonal polynomial solutions of a second order partial difference equation of hypergeometric type of two variables is done the pearson s systems for the orthogonality weight of the solutions and also for the difference derivatives of the solutions are presented the orthogonality property in subspaces is treated in detail which leads to an analog of the rodrigues type formula for orthogonal polynomials of two discrete variables a classification of the admissible equations as well as some examples related with bivariate hahn kravchuk meixner and charlier families and their algebraic and difference properties are explicitly given

an approach to the interpolation of nonuniformly spaced data in this paper we consider the spline in tension as an interpolatory tool that helps us avoid large projector norm tension parameters are computed in advance in such a way that the prescribed exponential decay of fundamental functions is assured further an interpolation error bound for a smooth function is produced this bound depends on the choice of tension parameters and is of order 4 for small tension values but reduces to 2 as parameters grow to infinity

new steps on sobolev orthogonality in two variables sobolev orthogonal polynomials in two variables are defined via inner products involving gradients such a kind of inner product appears in connection with several physical and technical problems matrix second order partial differential equations satisfied by sobolev orthogonal polynomials are studied in particular we explore the connection between the coefficients of the second order partial differential operator and the moment functionals defining the sobolev inner product finally some old and new examples are given

a triangle based mixed finite element finite volume technique for modeling two phase flow through porous media triangle based discretization techniques offer great advantages relative to standard finite difference methods for the modeling of flow through geometrically complex geological features the purpose of this paper is to develop and apply a triangle based method for the modeling of two phase flow through porous formations the formulation includes the effects of gravity compressibility and capillary pressure the technique entails a triangle based mixed finite element method for solution of the variable coefficient parabolic pressure equation and a second order tvd type total variation diminishing finite volume scheme for solution of the essentially hyperbolic saturation equation the method is applied to a variety of example problems and is shown to perform very well on problems involving geometric complexity coupled with heterogeneous generally anisotropic permeability descriptions

structure relations for monic orthogonal polynomials in two discrete variables in this paper extensions of several relations linking differences of bivariate discrete orthogonal polynomials and polynomials themselves are given by using an appropriate vector matrix notation three term recurrence relations are presented for the partial differences of the monic polynomial solutions of admissible second order partial difference equation of hypergeometric type structure relations difference representations as well as lowering and raising operators are obtained finally expressions for all matrix coefficients appearing in these finite type relations are explicitly presented for a finite set of hahn and kravchuk orthogonal polynomials

bivariate raising and lowering differential operators for eigenfunctions of a 2d fourier transform we define a two dimensional 2d fourier transform that self reproduces a one parameter family of bivariate hermite functions these are eigenfunctions of a hamiltonian differential operator of second order whose exponential is that transform we find explicit forms of the bivariate raising and lowering partial differential operators of first degree for the eigenfunctions of this 2d fourier transform

on koornwinder classical orthogonal polynomials in two variables in 1975 tom koornwinder studied examples of two variable analogues of the jacobi polynomials in two variables those orthogonal polynomials are eigenfunctions of two commuting and algebraically independent partial differential operators some of these examples are well known classical orthogonal polynomials in two variables such as orthogonal polynomials on the unit ball on the simplex or the tensor product of jacobi polynomials in one variable but the remaining cases are not considered classical by other authors the definition of classical orthogonal polynomials considered in this work provides a different perspective on the subject we analyze in detail koornwinder polynomials and using the koornwinder tools new examples of orthogonal polynomials in two variables are given

extensions of some results of p humbert on bezout s identity for classical orthogonal polynomials in this paper the bezout s identity is analyzed in the context of classical orthogonal polynomials solution of a second order differential equation of hypergeometric type differential equations relation with the starting family as well as recurrence relations and explicit representations are given for the bezout s pair extensions to classical orthogonal polynomials of a discrete variable and their q analogues are also presented applications of these results for the representation of the second kind functions are given