automatic mesh generation and adaptation by using contours a general and efficient remeshing algorithm is presented for the discretization of arbitrary planar domains into triangular elements in consistency with the given node spacing funstion the contour lines of the node spacing function at suitable calculated levels provide the natural lines of division of the problem domain into subregions where finite elements meshes of different element sizes zre generated using the available general purpose mesh generators

solving boundary value problems integral and integro differential equations using gegenbauer integration matrices we introduce a hybrid gegenbauer ultraspherical integration method hgim for solving boundary value problems bvps integral and integro differential equations the proposed approach recasts the original problems into their integral formulations which are then discretized into linear systems of algebraic equations using gegenbauer integration matrices gims the resulting linear systems are well conditioned and can be easily solved using standard linear system solvers a study on the error bounds of the proposed method is presented and the spectral convergence is proven for two point bvps tpbvps comparisons with other competitive methods in the recent literature are included the proposed method results in an efficient algorithm and spectral accuracy is verified using eight test examples addressing the aforementioned classes of problems the proposed method can be applied on a broad range of mathematical problems while producing highly accurate results the developed numerical scheme provides a viable alternative to other solution methods when high order approximations are required using only a relatively small number of solution nodes

on the coefficients of integrated expansions of bessel polynomials a new formula expressing explicitly the integrals of bessel polynomials of any degree and for any order in terms of the bessel polynomials themselves is proved another new explicit formula relating the bessel coefficients of an expansion for infinitely differentiable function that has been integrated an arbitrary number of times in terms of the coefficients of the original expansion of the function is also established an application of these formulae for solving ordinary differential equations with varying coefficients is discussed

on mass response functions field transport of reactive solute species is investigated through a class of stochastic models here termed mass response functions mrfs which incorporate simplified concepts of chemical physical nonequilibrium kinetics in the formulation of transport by travel time distributions mrfs are probability density functions pdfs associated with solute particles travel time within transport volumes the theory hinges on recent advances in modeling transport of solutes in groundwater and in basin scale transport volumes and links the approaches of surface hydrologists with recent subsurface transport models the relationship between mrfs and the theory of solute transport by continuous motions is investigated it is found that mrfs extend the basic formulation of transport of inert solutes to a particular case of sorption process the relationship between mrfs and the basic differential convection dispersion equation incorporating linear sorption is also investigated it is found here that not only are transfer functions of solutes consistent with any mechanistic three dimensional 3 d model of convection dispersion but also that they are under limit conditions the product of the travel time distribution of the carrier flow with a bounded continuous function the latter is the solution to an initial value problem which results from solving the general 3 d differential equations of convection dispersion with sorption under some simplifying assumptions and formally coincides with the resident concentration included as an assumption in the original mrf formulation travel time distributions and mrfs underlain by statistical constraints rather than by dynamical models are proposed non gaussian distributions are studied by statistical mechanical tools and are found to represent the norm rather than the exception in this formulation of transport of reactive solutes the concepts are applied to a field study and are shown to yield reliable models of solute migration in nonpoint pollution problems

improved condition number for spectral methods introduction d une methode spectrale avec un nombre de conditionnement o n 2 n etant le degre maximum des polynomes

statistical theory of groundwater flow and transport pore to laboratory laboratory to formation and formation to regional scale the statistical approach has been applied increasingly to groundwater flow problems in the last decade as is illustrated in figure 3 by the cumulative number of articles published in this field in water resources reseach this development has been motivated by the recognition of the fact that porous formations are heterogeneous i e with properties which vary in an irregular manner in space flow domains are characterized by the length scale l of their spatial extent and three such scales of a fundamental nature are introduced the laboratory the local and the regional scale heterogeneity is characterized by the spatial correlation scale i of the property of interest the three scales corresponding to the above ones being the pore scale the log hydraulic conductivity and the log transmissivity integral scales the medium properties and related flow variables are regarded as random space functions which satisfy two basic requirements they enjoy some type of stationarity and i l an additional scale d is the measurement or computational scale characterizing the size of the measurement device of a flow variable or the element over which the variable is averaged for computational purposes in both cases the interest resides in the space average of the flow variable over a volume or area of length scale d the primary aim of the theory of flow and transport through porous media is to determine the statistical moments of the space averaged variables given the statistical structure of the spatially variable property the main objective of the study is to show that flow and transport problems at the three fundamental scales can be treated by a unified statistical approach along this line the specific aspects of each scale are examined separately and areas of interest for future research are indicated in the concluding remarks it is submitted that the statistical approach to groundwater flow has become a comprehensive theory beyond the stage of an ad hoc modeling technique

on the coefficients of integrated expansions and integrals of ultraspherical polynomials and their applications for solving differential equations an analytical formula expressing the ultraspherical coefficients of an expansion for an infinitely differentiable function that has been integrated an arbitrary number of times in terms of the coefficients of the original expansion of the function is stated in a more compact form and proved in a simpler way than the formula suggested by phillips and karageorghis 27 1990 823 a new formula expressing explicitly the integrals of ultraspherical polynomials of any degree that has been integrated an arbitrary number of times of ultraspherical polynomials is given the tensor product of ultraspherical polynomials is used to approximate a function of more than one variable formulae expressing the coefficients of differentiated expansions of double and triple ultraspherical polynomials in terms of the original expansion are stated and proved some applications of how to use ultraspherical polynomials for solving ordinary and partial differential equations are described

automatic generation of anisotropic quadrilateral meshes on three dimensional surfaces using metric specifications a new algorithm for constructing full quadrilateral anisotropic meshes on 3d surfaces is proposed in this paper the proposed method is based on the advancing front and the systemic merging techniques full quadrilateral meshes are constructed by systemically converting triangular elements in the background meshes into quadrilateral elements by using the metric specifications to describe the element characteristics the proposed algorithm is applicable to convert both isotropic and anisotropic triangular meshes into full quadrilateral meshes special techniques for generating anisotropic quadrilaterals such as new selection criteria of base segment for merging new approaches for the modifications of the background mesh and construction of quadrilateral elements are investigated and proposed in this study since the final quadrilateral mesh is constructed from a background triangular mesh and the merging procedure is carried out in the parametric space the mesh generator is robust and no expensive geometrical computation that is commonly associated with direct quadrilateral mesh generation schemes is needed copyright 2002 john wiley sons ltd

three dimensional unstructured mesh generation part 1 fundamental aspects of triangulation and point creation abstract the present paper introduces an alternative approach for delaunay triangulation in which the triangulation is mapped from an equivalent convex hull in a higher dimension furthermore some fundamental aspects of a point creation algorithm for unstructured mesh generation have been addressed based on numerical experiments a point spacing tensor and point insertion criterion have been introduced aiming to deal with anisotropic meshes

a new finite point generation scheme using metric specifications summary a new approach to generate finite point meshes on 2d flat surface and any bi variate parametric surfaces is suggested it can be used to generate boundary conforming anisotropic point meshes with node spacing compatible with the metric specifications defined in a background point mesh in contrast to many automatic mesh generation schemes the advancing front concept is abandoned in the present method a few simple basic operations including boundary offsetting node insertion and node deletion are used instead the point mesh generation scheme is initialized by a boundary offsetting procedure the point mesh quality is then improved by node insertion and deletion such that optimally spaced nodes will fill up the entire problem domain in additional to the point mesh generation scheme a new way to define the connectivity of a point mesh is also suggested furthermore based on the connectivity information a new scheme to perform smoothing for a point mesh is proposed to improve the node spacing quality of the mesh timing shows that due to the simple node insertion and deletion operations the generation speed of the new scheme is nearly ten times faster than a similar advancing front mesh

dual contouring for domains with topology ambiguity this paper describes an automatic and robust approach to generate quality triangular meshes for complicated domains with topology ambiguity in previous works we developed an octree based dual contouring dc method to construct triangular meshes for complicated domains however topology ambiguity exists and causes non conformal meshes in this study we discuss all possible topology configurations and develop an extension of dc which guarantees the correct topology we first generate one base mesh with the previous dc method then we analyze all the octree leaf cells and categorize them into 31 topology groups in order to discriminate these cells we compute the values of their face and body saddle points based on a tri linear representation inside the cells knowing the correct categorization we are able to modify the base mesh and introduce more minimizer points within the same cell with these minimizer points we update the mesh connectivities to preserve the correct topology finally we use a laplacian smoothing technique to improve the mesh quality our main contribution is the topology categorization and mesh modification we have applied our algorithm to three complicated domains and obtained good results

adaptive triangulation of evolving closed or open surfaces by the advancing front method the advancing front method is adapted to describe the evolution of open or closed three dimensional surfaces in terms of an unstructured grid consisting of quadratic triangular elements in science and engineering applications the surface may be identified with a material interface a free boundary or a moving front in the numerical method the geometrical properties of the surface and the coordinates of the triangle vertices are computed either in terms of available analytical expressions or by means of interpolation through an underlying coarse grid the shape and size of the curved triangular elements are determined by the maximum magnitude of the mean or directional local surface curvature two algorithms are implemented the first one for simply connected open surfaces bounded by closed lines and for closed surface with plane symmetry and the second for closed surfaces in the case of open surfaces the discretization front advances from a one dimensional boundary grid that traces the bounding curve the boundary grid is generated either by a one dimensional version of the advancing front method or by requiring a set of criteria based on local line representation with circular arcs in the case of closed surfaces the discretization front emanates from the point of the maximum mean curvature in both cases the algorithm also interpolates to generate the values of surface geometrical or physical variables such as temperature or concentration of a surface active agent the method was tested by following the motion of several passive and active surfaces evolving under the action of specified fields of flow while performing occasional regridding to ensure adequate spatial resolution in one test the large deformation of a viscous drop subjected to an infinite simple shear flow at vanishing reynolds number was computed into the regime where a cigar like shape is established thereby extending previous numerical computations for small and moderate deformations and reproducing experimentally observed shapes overall the adaptive front method emerges as an important tool in numerical studies of free boundaries or moving fronts and should be useful in a broad range of applications

delaunay s mesh of a convex polyhedron in dimension d application to arbitrary polyhedra this paper presents a method for creating a delaunay triangulation connected to a set of specified points the theoretical aspect is recalled for an arbitrary dimension and the method is discussed in order to derive a practical approach valid for dimensions 2 and 3 which is simple robust and well adapted to computation convex polyhedral and arbitrary polyhedral situations are introduced

the advancing front mesh generation method revisited this paper presents a mesh generation method of the advancing front type which is designed in such a way that the well known difficulties of the classical advancing front method are not present the retained solution consists of using the first steps of a voronoi delaunay method to construct a background mesh which is then used to govern the algorithm the two dimensional case is considered in detail and possible extensions to adaption problems and three dimensions are indicated

kinematics of age mixing in advection dispersion models this paper investigates age mixing processes arising in advection dispersion models where large scale travel and residence time distributions can be explicitly calculated based on the underlying velocity field in particular we analyze spatially integrated age mixing dynamics by comparing the age distributions of the storage and of the outflow s the relevance of the work lies in the impact of age mixing dynamics on the shape of travel time distributions ttds which ultimately control the long term memory of catchment transport processes we set up a theoretical framework that bridges previous lagrangian and eulerian water age theories in heterogeneous media the framework allows for the analysis of the dynamical connection between water age distributions in large scale volume and flux samples theoretical advances are then illustrated through the application to a finite one dimensional domain with constant advection and dispersion coefficient therein we analyze the type of mixing emerging for different peclet numbers and diverse spatiotemporal patterns of solute input we find that in spite of the enhanced nonstationarity of ttds the type of mixing is markedly invariant moreover for relatively low peclet numbers the different ages available in the control volume are systematically removed from the domain at a rate nearly proportional to their relative abundance random sampling emerging large scale patterns of age mixing yield theoretical and practical implications for watershed hydrology where ttds can be used to infer general patterns of catchment response across scales

isogeometric analysis suitable trivariate nurbs models from standard b rep models abstract this paper presents an effective method to automatically construct trivariate spline models of complicated geometry and arbitrary topology required for nurbs based isogeometric analysis the input is a triangulation of the solid model s boundary the boundary surface is decomposed into a set of cuboids in two steps pants decomposition and cuboid decomposition the novelty of our method is the pants to cuboids decomposition algorithm the algorithm is completely automatic and very robust even for low quality and noisy meshes the set of cuboids composes a generalized polycube approximating very roughly the input model s geometry while faithfully replicating its topology due to its regular structure the polycube is suitable for serving as the parametric domain for the tensor product spline representation based on a discrete harmonic mapping between the cuboids boundary and the polycube s boundary surface and volume parameterizations are generated in order to fit the trivariate splines the efficiency and the robustness of the proposed approach are illustrated by several examples

on the coefficients of differentiated expansions and derivatives of jacobi polynomials a formula expressing explicitly the derivatives of jacobi polynomials of any degree and for any order in terms of the jacobi polynomials themselves is proved another explicit formula which expresses the jacobi expansion coefficients of a general order derivative of an infinitely differentiable function in terms of its original jacobi coefficients is also given the results for the special case of ultraspherical polynomials are considered the results for chebyshev polynomials of the first and second kinds and for legendre polynomials are also noted an application of how to use jacobi polynomials for solving ordinary and partial differential equations is described

on the construction of recurrence relations for the expansion and connection coefficients in series of jacobi polynomials formulae expressing explicitly the jacobi coefficients of a general order derivative integral of an infinitely differentiable function in terms of its original expansion coefficients and formulae for the derivatives integrals of jacobi polynomials in terms of jacobi polynomials themselves are stated a formula for the jacobi coefficients of the moments of one single jacobi polynomial of certain degree is proved another formula for the jacobi coefficients of the moments of a general order derivative of an infinitely differentiable function in terms of its original expanded coefficients is also given a simple approach in order to construct and solve recursively for the connection coefficients between jacobi jacobi polynomials is described explicit formulae for these coefficients between ultraspherical and jacobi polynomials are deduced of which the chebyshev polynomials of the first and second kinds and legendre polynomials are important special cases two analytical formulae for the connection coefficients between laguerre jacobi and hermite jacobi are developed

on semi infinite spectral elements for poisson problems with re entrant boundary singularities abstract a spectral element method is described which enables poisson problems defined in irregular infinite domains to be solved as a set of coupled problems over semi infinite rectangular regions two choices of trial functions are considered namely the eigenfunctions of the differential operator and chebyshev polynomials the coefficients in the series expansions are obtained by imposing weak c1 matching conditions across element interfaces singularities at re entrant corners are treated by a post processing technique which makes use of the known asymptotic behaviour of the solution at the singular point accurate approximations are obtained with few degrees of freedom

a quasi linear theory of non fickian and fickian subsurface dispersion 1 theoretical analysis with application to isotropic media a theory is presented which accounts for nonlinearity caused by the deviation of plume particles from their mean trajectory in three dimensional statistically homogeneous but anisotropic porous media under an exponential covariance of log hydraulic conductivities existing linear theories predict that in the absence of local dispersion transverse dispersivities tend asymptotically to zero as fickian conditions are reached according to our new quasi linear theory these dispersivities ascend to peak values and then diminish gradually toward nonzero fickian asymptotes which are proportional to y4 when the log hydraulic conductivity variance y2 is much less than 1 all existing theories agree that in isotropic media the asymptotic longitudinal dispersivity is proportional to y2 when y2 1 and all are nominally restricted to mildly heterogeneous media in which this inequality is satisfied however the quasi linear theory appears to be less prone to error than linear theories when extended to strongly heterogeneous media because it deals with the above nonlinearity without formally limiting y2 it predicts that when y 1 in isotropic media both the longitudinal and transverse dispersivities ascend monotonically toward fickian asymptotes proportional to y

jacobi spectral galerkin method for the integrated forms of second order differential equations abstract efficiently direct solvers based on the jacobi galerkin method for the integrated forms of second order elliptic equations in one and two space variables are presented they are based on appropriate base functions for the galerkin formulation which lead to discrete systems with specially structured matrices that can be efficiently inverted the homogeneous dirichlet boundary conditions are satisfied exactly by expanding the unknown variable into a polynomial basis of functions which are built upon the jacobi polynomials the direct solution algorithm here developed for the homogeneous dirichlet problem in two dimensions relies upon a diagonalization process numerical results indicating the high accuracy and effectiveness of these algorithms are presented

a note on the chebyshev coefficients of the general order derivative of an infinitely differentiable function abstract a formula expressing the chebyshev coefficients of the general order derivative of an infinitely differentiable function in terms of its chebyshev coefficients is suggested

tetrahedral meshes from planar cross sections in biomedicine many three dimensional 3d objects are sampled in terms of slices such as computed tomography ct magnetic resonance imaging mri and ultrasound imaging it is often necessary to construct surface meshes from the cross sections for visualization and thereafter construct tetrahedra for the solid bounded by the surface meshes for the purpose of finite element analysis in ref 1 c bajaj e coyle and k lin graphical models and image processing 58 6 1996 524 543 we provided a solution to the construction of a surface triangular mesh from planar section contours here we provide an approach to tetrahedralize the solid region bounded by planar contours and the surface mesh it is a difficult task because the solid can be of high genus several through holes as well as have complicated branching regions we develop an algorithm to effectively reduce the solid into prismatoids and provide an approach to tetrahedralize the prismatoids our tetrahedralization approach is similar to the advancing front technique aft for its flexible control of mesh quality the main criticism of aft is that the remaining interior may be badly shaped or even untetrahedralizable the emphasis of our prismatoid tetrahedralization approach is on the characterization and prevention of untetrahedralizable parts ruppert and seidel j ruppert r seidel on the difficulty of tetrahedralizing three dimensional non convex polyhedra in proceedings 5th annual acm symposium comput geom 1989 p 380 392 have shown that the problem of deciding whether a polyhedron is tetrahedralizable without adding steiner points is np complete we characterize this problem under certain constraints and design one rule to reduce the chance of generating untetrahedralizable shapes the characterization also leads to the classification of two common untetrahedralizable categories which can be better processed if they do occur

a new matrix method for solving two dimensional time dependent diffusion equations with dirichlet boundary conditions abstract this paper is devoted to develop a new matrix scheme for solving two dimensional time dependent diffusion equations with dirichlet boundary conditions we first transform these equations into equivalent integro partial differential equations pdes such these integro pdes contain both of the initial and boundary conditions and can be solved numerically in a more appropriate manner subsequently all the existing known and unknown functions in the latter equations are approximated by bernoulli polynomials and operational matrices of differentiation and integration together with the completeness of these polynomials can be used to reduce the integro pdes into the associated algebraic generalized sylvester equations for solving these algebraic equations an efficient krylov subspace iterative method i e bicgstab is implemented two numerical examples are given to demonstrate the efficiency accuracy and versatility of the proposed method

the ultraspherical coefficients of the moments of a general order derivative of an infinitely differentiable function abstract a formula for the ultraspherical coefficients of the moments of one single ultraspherical polynomial of certain degree is given formulae for the ultraspherical coefficients of the moments of a general order derivative of an infinitely differentiable function in terms of its ultraspherical coefficients are also obtained the corresponding formulae for the important special cases of chebyshev polynomials of the first and second kinds and of legendre polynomials are deduced two interesting numerical applications of how to use these formulae for solving ordinary differential equations with varying coefficients by reducing them to recurrence relations of lowest order in the ultraspherical expansion coefficients in the sense of lewanowicz are discussed comparisons with the results obtained by optimal algorithm of lewanowicz 1976 are also made

chebyshev expansion method for solving second and fourth order elliptic equations in this paper we introduce a new spectral method based on chebyshev polynomials for solving second and fourth order elliptic equations moreover the suggested method is applicable for a wide area of differential equations an explicit formula for the chebyshev polynomials in terms of arbitrary order of their derivatives is presented also a formula for the successive integration of chebyshev polynomials in terms of chebyshev polynomials is proved numerical results indicate that the suggested method is significantly more accurate than that based on the chebyshev tau method the present results are in satisfactory agreement with the exact solutions

jacobi spectral method for differential equations with rough asymptotic behaviors at infinity abstract in this paper we change differential equations on the half line to certain singular problems on finite interval and then use a specific jacobi approximation for solving the resulting problems numerically theoretical analysis and numerical results demonstrate the efficiency of this method

galerkin spectral method for the vorticity and stream function equations a galerkin legendre spectral method for the solution of the vorticity and stream function equations in uncoupled form under no slip conditions in a square domain is presented which fully exploits the separation of variables in the two elliptic problems benefits from a nonsingular influence matrix and is able to solve the singular driven cavity problem modulo gibbs phenomenon even without regularizing the boundary condition at the corners

a new approach to the development of automatic quadrilateral mesh generation a methodology for the automatic mesh generation of quadrilateral elements is proposed the methodology is based on the facts that i a region can always be subdivided entirely into quadrilaterals if the polygon which forms the boundary of the region has an even number of sides ii a quadrilateral can be formed by two triangles which share a common side by taking advantage of the techniques developed in an existing automatic triangular mesh generator quadrilateral meshes can be generated according to the distribution of user specified element size over the domain of interest in particular an automatic quadrilateral mesh generator based on the advancing front technique for triangular mesh generation1 is described techniques to improve the quality of the generated quadrilateral meshes are introduced the robustness and versatility of the mesh generator are demonstrated by a variety of examples

finite element euler computations in three dimensions an explicit finite element solution procedure for the three dimensional euler equations is presented the solution domain is automatically meshed using a tetrahedral mesh generator which is an extension of our previous two dimensional work several examples are included to illustrate the performance of the generator and solver an adaptive mesh regeneration procedure is used for the first time in three dimensions

spectral methods with sparse matrices spectral methods employ global polynomials for approximation hence they give very accurate approximations for smooth solutions unfortunately for dirichlet problems the matrices involved are dense and have condition numbers growing aso n 4 for polynomials of degree n in each variable we propose a new spectral method for the helmholtz equation with a symmetric and sparse matrix whose condition number grows only aso n 2 certain algebraic spectral multigrid methods can be efficiently used for solving the resulting system numerical results are presented which show that we have probably found the most effective solver for spectral systems

efficient spectral galerkin method ii direct solvers of second and fourth order equations using chebyshev polynomials efficient direct solvers based on the chebyshev galerkin methods for second and fourth order equations are presented they are based on appropriate base functions for the galerkin formulation that lead to discrete systems with special structured matrices that can be efficiently inverted numerical results indicate that the direct solvers presented in this paper are significantly more accurate and efficient than those based on the chebyshev tau method

efficient spectral petrov galerkin methods for the integrated forms of third and fifth order elliptic differential equations using general parameters generalized jacobi polynomials abstract this article analyzes some algorithms for solving numerically the integrated forms of third and fifth order differential equations in one variable subject to homogenous and nonhomogeneous boundary conditions using a dual petrov galerkin method two new families of general parameters generalized jacobi polynomials are introduced and used for this purpose numerical results indicating the high accuracy and effectiveness of the proposed algorithms are presented

mathematical analysis and calculation of molecular surfaces in this article we derive a complete characterization of the solvent excluded surface ses for molecular systems including a complete characterization of singularities of the surface the theory is based on an implicit representation of the ses which in turn is based on the signed distance function to the solvent accessible surface sas all proofs are constructive so that the theory allows for efficient algorithms in order to compute the area of the ses and the volume of the ses cavity or to visualize the surface further we propose to refine the notion of sas and ses in order to take inner holes in a solute molecule into account or not

an adaptive method for computing invariant manifolds in non autonomous three dimensional dynamical systems abstract we present a computational method for determining the geometry of a class of three dimensional invariant manifolds in non autonomous aperiodically time dependent dynamical systems the presented approach can be also applied to analyse the geometry of 3d invariant manifolds in three dimensional time dependent fluid flows the invariance property of such manifolds requires that at any fixed time they are given by surfaces in r 3 we focus on a class of manifolds whose instantaneous geometry is given by orientable surfaces embedded in r 3 the presented technique can be employed in particular to compute codimension one invariant stable and unstable manifolds of hyperbolic trajectories in 3d non autonomous dynamical systems which are crucial in the lagrangian transport analysis the same approach can also be used to determine evolution of an orientable material surface in a fluid flow these developments represent the first step towards a non trivial 3d extension of the so called lobe dynamics a geometric invariant manifold based framework which has been very successful in the analysis of lagrangian transport in unsteady two dimensional fluid flows in the developed algorithm the instantaneous geometry of an invariant manifold is represented by an adaptively evolving triangular mesh with piecewise c 2 interpolating functions the method employs an automatic mesh refinement which is coupled with adaptive vertex redistribution a variant of the advancing front technique is used for remeshing whenever necessary such an approach allows for computationally efficient determination of highly convoluted evolving geometry of codimension one invariant manifolds in unsteady three dimensional flows we show that the developed method is capable of providing detailed information on the evolving lagrangian flow structure in three dimensions over long periods of time which is crucial for a meaningful 3d transport analysis

on the dagan model of solute transport in groundwater foundational aspects the convection dispersion equation cde developed by dagan 1984 to predict the ensemble average concentration of a conservative solute in groundwater has been used successfully to interpret recent tracer experiments in a sand aquifer at the borden site in canada this successful application encouraged further investigation of the dagan model in respect to its physical basis and predictive characteristics it is shown in the present paper which deals with the first topic that the dagan model cde can be derived through an extension of the cumulant expansion technique applied previously by chu and sposito 1980 to develop a mean cde for solute transport in homogeneous porous media this technique also leads directly to the mean cde developed by gelhar and axness 1983 as an asymptotic large time limit and to model dispersion coefficients derived by mather on and de marsily 1980 and guven and moiz 1986 the dagan cde then is considered in detail in respect to which conceptualization of the solute concentration resident or flux the model may utilize in predictive applications general mathematical expressions relating the two conceptualizations are derived for an arbitrary solute transport problem and then are applied to the dagan model for point and prism source inputs finally the dagan model is interpreted physically using data from the recent tracer experiments of roberts and mackay 1986 at the borden site it is shown that model predictions of resident and flux concentrations are numerically indistinguishable on any time scale over which field solute concentration measurements typically are made the model prediction of dispersion coefficients on the other hand leads to dramatic differences in predicted plume behavior depending on whether finite time expressions or their asymptotic limits are used according to the dagan model the asymptotic transverse dispersion coefficient cannot replace the time dependent coefficient on any realistic time scale for solute movement at the borden site

stochastic modeling of groundwater flow by unconditional and conditional probabilities 2 the solute transport the problem of solute transport in a heterogeneous formation whose transmissivity or hydraulic conductivity are subject to uncertainty is studied for two and three dimensional flows approximate closed form solutions are derived for the case of a solute pulse in an average uniform flow through a formation of unconditional stationary random transmissivity the solute concentration regarded as a random variable is determined in terms of its expectation and variance and is found to be subject to a high degree of uncertainty the uncertainty is greatly reduced if the solute input zone is large compared to the transmissivity integral scale in any case the concentration expectation does not obey a diffusion type equation in the case of two dimensional flows unless the solute body has traveled a distance larger than a few tens transmissivity integral scales this distance may be exceedingly large in many conceivable applications

a spectral collocation method based on integrated chebyshev polynomials for two dimensional biharmonic boundary value problems this paper reports a new spectral collocation method for numerically solving two dimensional biharmonic boundary value problems the construction of the chebyshev approximations is based on integration rather than conventional differentiation this use of integration allows i the imposition of the governing equation at the whole set of grid points including the boundary points and ii the straightforward implementation of multiple boundary conditions the performance of the proposed method is investigated by considering several biharmonic problems of first and second kinds more accurate results and higher convergence rates are achieved than with conventional differential methods

spectral multigrid methods for elliptic equations ii a detailed description of spectral multigrid methods is provided this includes the interpolation and coarse grid operators for both periodic and dirichlet problems the spectral methods for periodic problems use fourier series and those for dirichlet problems are based upon chebyshev polynomials an improved preconditioning for dirichlet problems is given numerical examples and practical advice are included

a note on fractal channel networks this paper studies the relation between the structure of river networks and the features of their geomorphologic hydrologic response a mathematical formulation of connectivity of a drainage network is proposed to relate contributing areas and the network geometry in view of the connectivity conjecture horton s bifurcation ratio r b tends for high values of strahler s order omega of the basin to the area ratio r a and horton s length ratio r l equals in the limit the single order contributing area ratio r a the relevance of these arguments is examined by reference to data from real basins well known empirical results from the geomorphological literature melton s law hack s relation moon s conjecture are viewed as a consequence of connectivity it is found that in hortonian networks the time evolution of contributing areas exhibits a multifractal behavior generated by a multiplicative process of parameter 1 r b the application of the method of the most probable distribution in view of connectivity contributes new inroads toward a general formulation of the geomorphologic unit hydrograph in particular generalizing its width function formulation a quantitative example of multifractal hydrologic response of idealized networks based on peano s construct for which r b r a 4 r l 2 closes the paper

adaptive error control in multi physical thin structure mems fe simulation simulating the physical behavior of thin multi layered structures accurately is central to micro electro mechanical systems mems cad we have produced an automatic method with which to simulate the structural response of multi layer plate and beam micro structures accurately and reliably the method also covers thermo mechanical and piezoelectric effects we use a kirchhoff love thin structure model implemented as a conforming argyris finite element suited for the calculation of thermo mechanical membrane and bending behavior and which is extended to simulate piezoelectric effects in thin structures for the first time a posteriori estimation is presented for such multi layered multiphysically active thin structures different sources of errors are identified and specified for several usecases the error analysis covers locally prestressed regions plate composition inhomogeneities geometrical singularities and singularities conditional upon the presence of source functions of various types together with a refinement strategy and a geometrical split pattern the efficiency of the method is demonstrated local mesh refinement guarantees the computation of the most accurate solution at a minimum of computational costs

efficient spectral galerkin method i direct solvers of second and fourth order equations using legendre polynomials this paper presents some efficient algorithms based on the legendre galerkin approximations for the direct solution of the second and fourth order elliptic equations the key to the efficiency of these algorithms is to construct appropriate base functions which lead to systems with sparse matrices for the discrete variational formulations the complexities of the algorithms are a small multiple of n d 1 operations for a d dimensional domain with n 1 d unknowns while the convergence rates of the algorithms are exponential for problems with smooth solutions in addition the algorithms can be effectively parallelized since the bottlenecks of the algorithms are matrix matrix multiplications

optimal gegenbauer quadrature over arbitrary integration nodes this paper treats definite integrations numerically using gegenbauer quadratures the novel numerical scheme introduces the idea of exploiting the strengths of the chebyshev legendre and gegenbauer polynomials through a unified approach and using a unique numerical quadrature in particular the developed numerical scheme employs the gegenbauer polynomials to achieve rapid rates of convergence of the quadrature for the small range of the spectral expansion terms for a large scale number of expansion terms the numerical quadrature has the advantage of converging to the optimal chebyshev and legendre quadratures in the l norm and l 2 norm respectively the key idea is to construct the gegenbauer quadrature through discretizations at some optimal sets of points of the gegenbauer gauss gg type in a certain optimality sense we show that the gegenbauer polynomial expansions can produce higher order approximations to the definite integrals 1 x if x dx of a smooth function f x c 1 1 for the small range by minimizing the quadrature error at each integration point x i through a pointwise approach the developed gegenbauer quadrature can be applied for approximating integrals with any arbitrary sets of integration nodes exact integrations are obtained for polynomials of any arbitrary degree n if the number of columns in the developed gegenbauer integration matrix gim is greater than or equal to n the error formula for the gegenbauer quadrature is derived moreover a study on the error bounds and the convergence rate shows that the optimal gegenbauer quadrature exhibits very rapid convergence rates faster than any finite power of the number of gegenbauer expansion terms two efficient computational algorithms are presented for optimally constructing the gegenbauer quadrature we illustrate the high order approximations of the optimal gegenbauer quadrature through extensive numerical experiments including comparisons with conventional chebyshev legendre and gegenbauer polynomial expansion methods the present method is broadly applicable and represents a strong addition to the arsenal of numerical quadrature methods

on the coefficients of integrated expansions of ultraspherical polynomials the problem of expressing the coefficients of an expansion of orthogonal polynomials that has been integrated an arbitrary number of times in terms of the coefficients of the original expansion is considered a formula is proved for the ultraspherical polynomials of which the chebyshev and legendre polynomials are important special cases

konvergenzaussagen fur kollokationsverfahren bei elliptischen randwertaufgaben we examine the convergence of collocation with polynomials for elliptic problems the general theory is imbedded in thel p theory from which known results of existence and regularity are adopted convergence results are deduced in the cases of periodic dirichlet and mixed boundary conditions it becomes obvious that the order of convergence essentially depends on the smoothness of the solution