bluuebunny/crossref_metadata_2025 huggingface dataset:
A finite element collocation method for variably saturated flows in porous media
Authors: Allen Myron B. , Murphy Carolyn
Published: August 2005

One common formulation of Richard's equation for variably saturated flows in porous media treats pressure head as the principal unknown and moisture content as a constitutive variable. Numerical approximations to this “head‐based” formulation often exhibit mass‐balance errors arising from inaccuracies in the temporal discretization. This article presents a finite‐element collocation scheme using a mass‐conserving formulation. The article also proposes a computable index of global mass balance.
A nondispersive and nondissipative numerical method for first‐order linear hyperbolic partial differential equations
Authors: Watanabe Yoichi
Published: August 2005

We propose a new numerical method for a solution of first‐order linear hyperbolic equations. The leap‐frog scheme is converted to a nondispersive scheme by introducing an adjustable constant in a fictitious absorption term. Then the erroneous decrease in th solution is eliminated by solving two equations equivalent to the original equation. The new scheme perfectly preserves the form of a discontinuous solution.
Superconvergent recovery of gradients on subdomains from piecewise linear finite‐element approximations
Authors: Wheeler M. F. , Whiteman J. R.
Published: August 2005

Engineers have been aware for some time of the phenomenon of superconvergence, whereby there exist (stress) points at which the accuracy of a finite‐element solution is superior to that of the approximation generally. This phenomenon has been treated in recent years by mathematicians who have proved, for certain two‐dimensional secondorder elliptic problems, superconvergent error estimates for retrieved finite‐element derivatives. These results have demanded high global regularity of the solutions of the bondary value problems. In this present article cut‐off functions are used to prove similar superconvergence results over interior subdomains. This allows superconvergence estimates to be derived for problems with solutions of low global regularity, particularly those involving singularities.
Open boundary conditions for hyperbolic equations
Authors: Lick Wilbert , Ziegler Kirk , Lick James
Published: August 2005

A previously developed general procedure for deriving accurate difference equations to describe conditions at open boundaries for hyperbolic equations is extended and further illustrated by means of several examples of practical importance. Problems include those with both incoming and outgoing waves at the boundary, the use of locally cylindrical and spherical wave approximations at each point of the boundary, and nonlinear wave propagation. Reflected waves in all cases are minimal and less than 10−2 of the incident wave.
An investigation of chaos in reaction‐diffusion equations
Authors: Wachholz James J. , Bruch John C.
Published: August 2005

The purpose of this article is to investigate graphically and numerically the topic of chaos in reaction‐diffusion equations. This article is based on the article by Mitchell and Bruch [1]. One‐ and two‐dimensional forms of the reaction‐diffusion equation are discretized using the explicit Euler finite difference scheme. Plots are presented to show the effect of bifurcation parameters on the difference equations. Varying these parameters produce single point, periodic, chaotic, intermittent, and divergent solutions.
Numerical implementation of the Sinc‐Galerkin method for second‐order hyperbolic equations
Authors: McArthur Kelly M. , Bowers Kenneth L. , Lund John
Published: August 2005

A fully Galerkin method in both space and time is developed for the second‐order, linear hyperbolic problem. Sinc basis functions are used and error bounds are given which show the exponential convergence rate of the method. The matrices necessary for the formulation of the discrete system are easily assembled. They require no numerical integrations (merely point evaluations) to be filled. The discrete problem is formulated in two different ways and solution techniques for each are described. Consideration of the two formulations is motivated by the computational architecture available. Each has advantages for the appropriate hardware. Numerical results reported show that if 2N + 1 basis functions are used then the exponential convergence rate \documentclass{article}\pagestyle{empty}\begin{document}$ 0\left[{\exp \left({- \kappa \sqrt N} \right)} \right] $\end{document}, κ > 0, is attained for both analytic and singular problems.
High‐order methods for elliptic equations with variable coefficients
Authors: Ananthakrishnaiah U. , Manohar R. , Stephenson J. W.
Published: August 2005

In this article, we give a simple method for developing finite difference schemes on a uniform square gird. We consider a general, two‐dimensional, second‐order, partial differential equation with variable coefficients. In the case of a nine‐point scheme, we obtain the known results of Young and Dauwalder in a fairly elegant fashion. We show how this can be extended to obtain fourth‐order schemes on thirteen points. We derive two such schemes which are attractive because they can be adapted quite easily bnto obtain formulas for gird points near the boundary. In addition to this, these formulas only require nine evaluations for the typical forcing function. Numerical examples are given to demonstrate the performance of one of the fourth‐order schemes.
Fourth‐order finite difference methods for three‐dimensional general linear elliptic problems with variable coefficients
Authors: Ananthakrishnaiah U. , Manohar R. , Stephenson J. W.
Published: August 2005

In this article, we give a simple method for deriving finite difference schemes on a uniform cubic grid. We consider a general, three‐dimensional, second‐order, linear, elliptic partial differential equation with variable coefficients. We derive two simple fourth‐order schemes. When the coefficients of the second‐order mixed derivatives are equal to zero, the fourth‐order scheme requires only 19 grid points. When the coefficients of the mixed derivatives are not equal to zero and the coefficients of Uxx, Uyy, and Uzz are equal, we require the 27 points of the cubic grid. Numerical examples are given to demonstrate the performance of the two schemes derived. There does not exist a fourth‐order scheme involving 27 grid points for the general case.
Hermitian operator methods for reaction‐diffusion equations
Authors: Ramos J. I.
Published: August 2005

A variety of time‐linearization, quasilinearization, operator‐splitting, and implicit techniques which use compact or Hermitian operators has been developed for and applied to one‐dimensional reaction‐diffusion equations. Compact operators are compared with second‐order accurate spatial approximations in order to assess the accuracy and efficiency of Hermitian techniques. It is shown that time‐linearization, quasilinearization, and implicit techniques which use compact operators are less accurate than second‐order accurate spatial discretizations if first‐order approximations are employed to evaluate the time derivatives. This is attributed to first‐order accurati temporal truncation errors. Compact operator techniques which use second‐order temporal approximations are found to be more accurate and efficient than second‐order accurate, in both space and time, algorithms. Quasilinearization methods are found to be more accurate than time‐linearization schemes. However, quasilinearization techniques are less efficient because they require the inversion of block tridiagonal matrices at each iteration. Some improvements in accuracy can be obtained by using partial quasilinearization and linearizing each equation with respect to the variable whose equation is being solved. Operator‐splitting methods which use compact differences to evaluate the diffusion operator were found to be less accurate than operator‐splitting procedures employ second‐order accurate spatial approximations. Comparisons among the methods presented in this paper are shown in terms of the L2‐norm errors and computed wave speeds for a variety of time steps and grid spacings: The numerical efficiency is assessed in terms of the CPU time required to achieve the same accuracy.
Difference methods for quasilinear parabolic systems from plasma physics
Authors: Von Finckenstein Karl Graf Finck
Published: August 2005

A class of nonlinear implicit one‐step difference methods for quasilinear strongly coupled parabolic systems in cylinderical symmetry is considerd. The main part of the article deals with proving convergence of both the semi‐discrete and the fully discrete approximations. For this purpose, bounds for the inverse difference operators have to be derived previously. This is possible subject to a condition which can be considered as a generalization of the concept “parabolic” to systems. Furthermore, a linearized scheme already used by Düchs some years ago turns out to be nothing but one iteration step of the Newton mehod for solving the complete system of nonlinear difference equations. A surprising improvement on the linearized scheme is achievd, if two or more Newton iterations are performed.
Superconvergent recovery of gradients on subdomains from piecewise linear finite‐element approximations
Authors: Wheeler M. F. , Whiteman J. R.
Published: August 2005

Engineers have been aware for some time of the phenomenon of superconvergence, whereby there exist (stress) points at which the accuracy of a finite‐element solution is superior to that of the approximation generally. This phenomenon has been treated in recent years by mathematicians who have proved, for certain two‐dimensional second‐order elliptic problems, superconvergent error estimates for retrieved finite‐element derivatives. These results have demanded high global regularity of the solutions of the boundary value problems. In this present article cut‐off functions are used to prove similar superconvergence results over interior subdomains. This allows superconvergence estimates to be derived for problems with solutions of low global regularity, particularly those involving singularities.
Numerical comparison of preconditionings for large sparse finite element problems
Authors: Gambolati G. , Pini G. , Zilli G.
Published: August 2005

A numerical study of the efficiency of the modified conjugate gradients (MCG) is performed using different preconditioning schemes. The MCG behavior is evaluated in connection with the solution of large linear sets of symmetric positive definite (p.d.) equations, arising from the finite element (f.e.) integration of partial differential equations of parabolic and elliptic type and the analysis of the leftmost eingenspectrum of the corresponding matrices. A simple incomplete Cholesky factorization ICCG(O) having the same sparsity pattern as the original problem is compared with a more complex technique ICAJ (Ψ) where the triangular factor is allowed to progressively fill in depending on a rejection parameter Ψ. The performance of the preconditioning algorithms is explored on finite element equations whose sizeNranges between 150 and 2300. The results show that an optimal Ψoptmay be found which minimizes the overall CPU time for the solution of both the linear system and the eigenproblem. The comparison indicates that ICAJ (Ψopt) is not significantly more efficient than ICCG(O), which therefore appears to be a simple, robust, and reliable method for the preconditioning of large sparse finite element models.
A regularization‐stabilization technique for nonlinear conservation equation computations
Authors: Carey G. F. , Jiang B. N. , Showalter R.
Published: August 2005

A regularization procedure for nonlinear conservation equations is introduced and demonstrated to have a stabilizing effect on the numerical solution of the associated approximate problem. Representative results for a least‐squares finite‐element method are given, and the numerical performance of the stabilization procedure explored. The effect of the regularization term is similar to a local numerical dissipation dependent on the numerical itegration time step.
Six combinations of the Ritz‐Galerkin and finite element methods for elliptic boundary value problems
Authors: Li Z. C. , Bui T. D.
Published: August 2005

Overall comparisons are made for six efficient combinations of the Ritz‐Galerkin and finite element methods for solving elliptic boundary value problems with singularities or interfaces. The comparisons are done by using theoretical analysis and numerical experiments. Significant relations among the six combinations are also found. A survey of the six combinations and their coupling strategies are given. These combinations are important not only for matching the Ritz‐Galerkin method and the finite element method but also for matching other numerical methods such as the Ritz‐Galerkin method and the finite difference method.
Numerical solution of the Fokker Planck equation using moving finite elements
Authors: Harrison Gray W.
Published: August 2005

Numerical solution of the Fokker Planck equation for the probability density function of a stochastic process by traditional finite difference or finite element methods produces erroneous oscillations and negative values whenever the drift is large compared to the diffusion. Upwinding schemes to eliminate the oscillations introduce false numerical diffusion because it is impossible to make the one step drift large enough to match the original equation without making the one step diffusion too large. A variation of the moving finite element method is presented that overcomes these difficulties by using basis functions that satisfy the drift part of the equation by moving along the trajectories of the deterministic dynamical system associated with the stochastic process. A Galerkin type method can then be used to find the coefficients in the remaining pure diffusion equation. Solutions of two test equations are presented to illustrate the effectiveness of the method.
On a zero equation model of turbulence
Authors: Turner James C.
Published: August 2005

In this paper we study a widely used zero equation model of turbulence. The governing equations are derived by applying to the incompressible Navier‐Stokes equations the Reynolds time averaging procedure. We achieve closure by employing the eddy viscosity concept. Using the Implicit Function Theorem we obtain an existence and uniquencess result. We also discuss the existence of nonsingular solutions. Finally, we present an algorithm for solving the modeled equations.
A new numerical approach for the advective‐diffusive transport equation
Authors: Celia Michael A. , Herrera Ismael , Bouloutas Efthimios , Kindred J. Scott
Published: August 2005

A new numerical solution procedure is presented for the one‐dimensional, transient advective‐diffusive transport equation. The new method applies Herrera's algebraic theory of numerical methods to the spatial derivatives to produce a semi‐discrete approximation. The semi‐discrete system is then solved by standard time marching algorithms. The algebraic theory, which involves careful choice of test functions in a weak form statement of the problem, leads to a numerical approximation that inherently accommodates different degrees of advection domination. Algorithms are presented that provide either nodal values of the unknown function or nodal values of both the function and its spatial derivative. Numerical solution of several test problems demonstrates the behavior of the method.
An approach combining the Ritz‐Galerkin and finite difference methods
Authors: Li Zi‐Cai
Published: August 2005

The nonconforming combination of Ritz‐Galerkin and finite difference methods is presented for solving elliptic boundary value problems with singularities. The Ritz‐Galerkin method is used in the subdomains including singularities, the finite difference method is used in the rest of the solution domain. Moreover, on the common boundary of two regions where two different methods are used, the continuity conditions are constrained only on the nodes of difference grids. Theoretical analysis and numerical experiments have shown that average errors of numerical solutions and their generalized derivatives can reach the convergence rate O(h2‐δ), where h is the mesh spacing of uniform difference grids, and δ is an arbitrarily small, positive number. This convergence rate is better than O(h), obtained by the nonconforming combination of the Ritz‐Galerkin and finite element methods.
Recovery of an unknown specific heat by means of overposed data
Authors: Pilant Michael , Rundell William
Published: August 2005

In this paper we consider a class of inverse problems in which an unknown function, c(.), is to be determined from a parabolic initial‐value problem, with overposed Dirichlet data along a portion of the boundary. A mapping between the overposed data and the unknown coefficient is obtained in the form of a singular integral equation. This is solved by iteration, and the resulting fixed point is shown to be the solution of the inverse problem. Sufficient conditions for convergence of this method, as well as an extension to the case of an unknown thermal conductivity, are given.
Numerical approaches for computation of fronts
Authors: Sheintuch Moshe
Published: August 2005

Moving fronts and pulses appear in many engineering applications like flame propagation and a falling liquid film. Standard computation methods are inappropriate since the problem is defined over an infinite domain and a steady‐state solution exists only for a certain front velocity. This work presents a transformation that converts the original problem into a boundary‐value problem within a finite domain, in a way that preserves the behavior at the boundaries. Good low‐order approximations can be obtained as demonstrated by two examples. In another approach, a central element of adjustable length is incorporated into a three‐element structure where the edge‐elements obey known asymptotic solutions. That yields multiplicity of travelling fronts in an infinite domain but it successfully approximates standing wave solutions in a finite domain. The approximate solutions are shown to obey the qualitative features known for the exact solutions, like asymptotic solutions or the bifurcation set–the boundary where a new solution emerges or disappears.
Pointwise superconvergence of recovered gradients for piecewise linear finite element approximations to problems of planar linear elasticity
Authors: Goodsell G. , Whiteman J. R.
Published: August 2005

In this paper we consider piecewise linear finite element approximations to the problem of planar linear elasticity, and present some new estimates for the pointwise (L∞) superconvergence of a recovered gradient function to the gradient of the true solution. This extends to linear elasticity the previous work of the present and other authors on L∞ results for Poisson problems, and at the same time, to the L∞ norm the previous L2 results of the authors for linear elasticity.
An analysis of the finite element method for natural convection problems
Authors: Boland J. , Layton W.
Published: August 2005

We derive stability properties and error estimates for the finite element method when used to approximate heat flow in a fluid enclosed by a solid medium. The coupled Navier Stokes system involves the Boussinesq equations in the fluid‐filled cavity linked through an interface with heat conduction in the solid enclosing the fluid. As we assume no extra regularity then can be shown to hold under mild restriction on the data (at least over a small time interval in R3), we focus primarily on low order finite element spaces.
Path integral numerical methods for evolution equations
Authors: Miranker Willard L. , Winkler Andrew
Published: August 2005

A universal method for representing solutions of the initial value problem ut = −2πiDu is presented for a large class of pseudo‐differential operators. The representations take the form of a path integral‐like operator. Three numerical approaches for evaluating these operators are treated. Practical interest in such universal methods depends on massively parallel computing systems.
A direct general finite difference method for two‐dimensional heat conduction problems
Authors: Dassle G. , Reall M.
Published: August 2005

A numerical method, based on general finite difference forms, is introduced and applied for solving heat conduction problems in two‐dimensional domains of any shape and compounded by subdomains with differing thermal properties.
Numerical solutions of some parabolic inverse problems
Authors: Cannon John R. , Yin Hong‐Ming
Published: August 2005

In this article we transform a large class of parabolic inverse problems into a nonclassical parabolic equation whose coefficients consist of trace type functionals of the solution and its derivatives subject to some initial and boundary conditions. For this nonclassical problem, we introduce a variational form by defining a new function. Both continuous and discrete Galerkin procedures are illustrated in this paper. The error estimates are also derived.
Generalized alternating‐direction collocation methods for parabolic equations. I. Spatially varying coefficients
Authors: Celia Michael A. , Pinder George F.
Published: August 2005

The alternating‐direction collocation (ADC) method can be formulated for general parabolic partial differential equations. This is done using a piecewise cubic Hermite trial space defined on a rectangular discretization. As in all alternating‐direction methods, the ADC algorithm produces errors that are additional to the standard discretization errors of multi‐dimensional collocation. These errors increase when the coefficients of the governing equation are spatially variable. Analysis of the additional errors leads to several correction schemes. Numerical results indicate that a variant on the Laplace‐modification procedure is an attractive choice as an improved ADC algorithm.
Generalized alternating‐direction collocation methods for parabolic equations. II. Transport equations with application to seawater intrusion problems
Authors: Celia Michael A. , Pinder George F.
Published: August 2005

The alternating‐direction collocation (ADC) method combines the attractive computational features of a collocation spatial approximation and an alternating‐direction time marching algorithm. The result is a very efficient solution procedure for parabolic partial differential equations. To date, the methodology has been formulated and demonstrated for second‐order parabolic equations with insignificant first‐order derivatives. However, when solving transport equations, significant first‐order advection components are likely to be present. Therefore, in this paper, the ADC method is formulated and analyzed for the transport equation. The presence of first‐order spatial derivatives leads to restrictions that are not present when only second‐order derivatives appear in the governing equation. However, the method still appears to be applicable to a wide variety of transport systems. A formulation of the ADC algorithm for the nonlinear system of equations that describes density‐dependent fluid flow and solute transport in porous media demonstrates this point. An example of seawater intrusion into coastal aquifers is solved to illustrate the applicability of the method.

An alternating‐direction collocation solution algorithm has been developed for the general transport equation. The procedure is analogous to that for the model parabolic equations considered by Celia and Pinder [2]. However, the presence of first‐order spatial derivatives requires special attention in the ADC formulation and application. With proper implementation, the ADC procedure effectively combines the efficient equation formulation inherent in the collocation method with the efficient equation solving characteristics of alternating‐direction time marching algorithms. To demonstrate the viability of the method for problems with complex velocity fields, the procedure was applied to the problem of density‐dependent flow and contaminant transport in groundwaters. A standard example of seawater intrusion into coastal aquifers was solved to illustrate the applicability of the method and to demonstrate its potential use in practical problems.
Generalized alternating‐direction collocation methods for parabolic equations. III. Nonrectangular domains
Authors: Celia Michael A. , Pinder George F.
Published: August 2005

The alternating‐direction collocation (ADC) method is an efficient numerical approximation technique for the solution of parabolic partial differential equations. However, to date the ADC method has only been developed for rectangular discretizations. With judicious combination of isoparametric coordinate transformations and an extended ADC approach, the ADC method can be formulated on general nonrectangular domains. This extends the applicability of the ADC method by allowing it to be employed on domains of more general geometry.
The effect of numerical integration in finite element methods for nonlinear parabolic equations
Authors: Chou So‐Hsiang , Li Qian
Published: August 2005

We consider the effect of numerical integration in finite element methods applied to a class of nonlinear parabolic equations and give some sufficient conditions to ensure that the order of convergence is unaltered in the presence of numerical integration. Optimal error estimates for the solution and its time derivative in the spaces L∞(H1) and L∞(L2) are established. Moreover, maximum norm error estimates are demonstrated.
Characteristic adaptive subdomain methods for reservoir flow problems
Authors: Dahle H. K. , Espedal M. S. , Ewing R. E. , Sævereid O.
Published: August 2005

The simulation of multiphase and multicomponent fluid flows often requires the solution of large, coupled systems of nonlinear partial differential equations of convection‐advection type. The equations are usually advection dominated with small but important local diffusive effects. An operator‐splitting technique is used to address these different phenomena. A modified method of characteristics is used to try to eliminate the nonsymmetry in the operators due to advection. Although this method does not completely symmetrize the problem with nonlinear, nonconvex flux functions, the remaining operator is almost symmetric and can be stabilized via Petrov‐Galerkin techniques with optimal or near optimal test functions. Effective localized test functions are descrbed. A coarse grid is defined to treat the slow variation of the fluid velocity away from fronts. A substructuring of this grid gives a proper frontal resolution via local patch grid refinement. Preconditioning techniques are presented to efficiently solve the resulting composite‐grid problem. Error estimates for the total approximation procedure are presented. Finally, certain two‐dimensional computations are described.
Influence matrix technique for the numerical spectral simulation of viscous incompressible flows
Authors: Phillips Timothy N. , Soliman Ibrahim M.
Published: August 2005

A spectral algorithm based on the influence matrix technique is desclibed for solving numerically the flow of incompressible viscous fluids. The algorithmic development is for both Newtonian and non‐Newtonian flows. To investigate the performance of the method several test problems are solved. Accurate results are obtained with relatively few degrees of freedom.
Prismatic versus tetrahedral elements in three‐dimensional finite element analyses of subsurface systems
Authors: Pini G.
Published: August 2005

A mesh of prismatic or tetrahedral elements automatically generated from an initial triangular grid is used to integrate 3‐D flow equation in space. Many numerical comparisons between these two models have been performed. The results show that integration with tetrahedrons is as accurate as integration with prisms but much more efficient. The CPU time of solution with prismatic elements is about three times greater than that required employing tetrahedral elements.
Fourth‐order difference methods for the system of 2D nonlinear elliptic partial differential equations
Authors: Jain M. K. , Jain R. K. , Mohanty R. K.
Published: August 2005

We present the fourth‐order finite difference methods for the system of 2D nonlinear elliptic equations using 9‐grid points on a square region R subject to Dirichlet boundary conditions. The method has been tested on viscous, incompressible 2D Navier‐Stokes equations. The numerical results show that the proposed methods produce accurate and oscillation‐free solutions for large Reynolds numbers.
Natural and postprocessed superconvergence in semilinear problems
Authors: Chow S.‐S. , Carey G. F. , Lazarov R. D.
Published: August 2005

Superconvergence error estimates are established for a class of semilinear problems defined by a linear elliptic operator with a nonlinear forcing term. The analysis is for rectangular biquadratic elements, and we prove superconvergence of the derivative components along associated lines through the Gauss points. Derivative postprocessing formula and formulas for integrals are also considered and similar superconvergence estimates proven.
Convergence of the contour dynamics method
Authors: Soler Juan
Published: August 2005

In this paper the contour dynamics method (CDM) is formulated and applied to the study of a two‐dimensional incompressible and inviscid fluid flow. A result concerning the convergence of this method is proved. The foundation of the CDM is based on the study of the dynamic interaction between contours enclosing vortex regions. The idea of the analysis is to show that the CDM can be considered as a vortex filament method. Hence, the convergence for the contour dynamics method is proved by using some results in the theory of the vortex method.
Discontinuous solutions to hyperbolic systems under operator splitting
Authors: Roe P. L.
Published: August 2005

Two‐dimensional systems of linear hyperbolic equations are studied with regard to their behavior under a solution strategy that in alternate time‐steps exactly solves the component one‐dimensional operators. The initial data is a step function across an oblique discontinuity. The manner in which this discontinuity breaks up under repeated applications of the split operator is analyzed, and it is shown that the split solution will fail to match the true solution in any case where the two operators do not share all their eigenvectors. The special case of the fluid flow equations is analyzed in more detail, and it is shown that arbitrary initial data gives rise to “pseudo acoustic waves” and a nonphysical stationary wave. The implications of these findings for the design of high‐resolution computing schemes are discussed.
Fourth‐order finite difference method for 2D parabolic partial differential equations with nonlinear first‐derivative terms
Authors: Jain M. K. , Jain R. K. , Mohanty R. K.
Published: August 2005

We attempt to obtain a two‐level implicit finite difference scheme using nine spatial grid points of O(k2 + kh2 + h4) for solving the 2D nonlinear parabolic partial differential equation v1uxx + v2uyy = f(x, y, t, u, ux, uy, u1) where v1 and v2 are positive constants, with Dirichlet boundary conditions. The method, when applied to a linear diffusion‐convection problem, is shown to be unconditionally stable. Computational efficiency and the results of numerical experiments are discussed.
Simple nonconforming quadrilateral Stokes element
Authors: Rannacher R. , Turek S.
Published: August 2005

A simple nonconforming quadrilateral Stokes element based on “rotated” multi‐linear shape functions is analyzed. On strongly nonuniform meshes the usual parametric version of this element suffers from a lack of consistency, while its nonparametric counterpart turns out to be convergent with optimal orders. This theoretical result is confirmed by numerical tests.
Variational approach to grid generation
Authors: Liao G.
Published: August 2005

Some mathematical aspects of the variational approach to grid generation are discussed. Existence and regularity results for 2D are presented.
Stability analysis of discrete generalized Stokes problems
Authors: Fortin M. , Pierre R.
Published: August 2005

In the mixed finite element approximation of Stokes‐like problems, stability considerations normally restrict the choice of the finite element subspaces to families satisfying the so‐called inf‐sup condition. However, in the case of the generalized Stokes problems that one encounters when solving either the nonstationary compressible or incompressible Navier–Stokes equations by means of some operator splitting, the contribution of the time derivative is adding a term in the continuity equations which, at least theoretically, leads to a problem that is always stable. On the other hand, the numerical solution of the discrete problems built with approximations not satisfying the inf‐sup condition are generally plagued with oscillations that look pretty much like the checkerboard phenomenon occurring in the classical context. To clarify this apparent contradiction, we undertook a theoretical study of the condition number of the linear problem involving the sensitive variable, which for such a saddle point problem is the dual problem. In the course of this study, we have exhibited the relationship between that condition number and the inf‐sup constant, thus providing a different point of view on the results of Brezzi and Babuska. Finally, we conducted a parametric study of the behavior of that condition number with respect to the mesh size, the viscosity, and the time step. That study showed that, even in this context, the use of “stable element” was adding to the quality of the approximation by reducing the condition number to an acceptable level.
Domain‐decomposition approach to local grid refinement in finite element collocation
Authors: Curran Mark C. , Allen Myron B.
Published: August 2005

Advection‐dominated flows occur widely in the transport of groundwater contaminants, the movements of fluids in enhanced oil recovery projects, and many other contexts. In numerical models of such flows, adaptive local grid refinement is a conceptually attractive approach for resolving the sharp fronts or layers that tend to characterize the solutions. However, this approach can be difficult to implement in practice. A domain decomposition method developed by Bramble, Ewing, Pasciak, and Schatz, known as the BEPS method, overcomes many of the difficulties. We demonstrate the applicability of BEPS ideas to finite element collocation on trial spaces of piecewise Hermite cubics. The resulting scheme allows one to refine selected parts of a spatial grid without destroying algebraic efficiencies associated with the original coarse grid. We apply the method to steady‐state problems with boundary and interior layers and a time‐dependent advection‐diffusion problem.
The primal mixed finite element method and the LBB condition
Authors: Robey Thomas H.
Published: August 2005

Spurious or kinematic modes have posed a major obstacle to the implementation of the mixed finite element method. This research shows that spurious modes resulting from the approximation spaces not satisfying the LBB condition do not prevent a well posed problem. When the LBB condition is not satisfied, the resulting matrix equations are singular. A direct solution method is presented for the efficient solution of the possibly singular equations. Orthogonal flux basis functions are introduced to simplify the problem. Then the solution procedure is based on nested domain decomposition. This solution procedure is shown to be competitive with direct solution methods for the displacement finite element method. Examples are included to demonstrate various aspects of the LBB condition and the solution procedure.
Error estimates for mixed finite element methods for nonlinear parabolic problems
Authors: Chou So‐Hsiang , Li Qian
Published: August 2005

We consider a class of mixed finite element methods for nonlinear parabolic problems over a plane domain. The finite element spaces taken are Raviart‐Thomas spaces of index k, k ⩾ 0. We obtain optimal order L2‐ and almost optimal order L∞‐error estimates for the finite element solution and order optimal L2‐error estimates for its gradient. We also derive the error estimates for the time derivatives of the solution. Our results extend those previously obtained by Johnson and Thomée for the corresponding linear problems with k ⩾ 1.
Numerical solution of the heat equation by infinite‐order ordinary differential equations
Authors: Hills Norman L. , Irwin John M. , Mulla Fuad
Published: August 2005

A numerical algorithm is given for solving a class of infinite‐order differential equations previously discussed by the authors. These equations yield solutions to certain time‐dependent boundary‐value problems for the heat equation. An extrapolation process is given for increasing the speed of convergence of the sequence generated by the method. © 1992 John Wiley & Sons, Inc.
Adaptive refinement procedure for the finite difference method
Authors: Dow John O. , Stevenson Ian
Published: August 2005

An adaptive refinement procedure consisting of a localized error estimator and a physically based approach to mesh refinement is developed for the finite difference method. The error estimator is a variation of a successful finite element error estimator. The errors are estimated by computing an error energy norm in terms of discontinuous and continuous stress fields formed from the finite difference results for plane stress problems. The error measure identifies regions of high error which are subsequently refined to improve the result. The local refinement procedure utilizes a recently developed approach for developing finite difference templates to produce a graduated mesh. The adaptive refinement procedure is demonstrated with a problem that contains a well‐defined singularity. The results are compared to finite element and uniformly refined finite difference results.
Fourth‐order finite difference method for three‐dimensional elliptic equations with nonlinear first‐derivative terms
Authors: Jain M. K. , Jain R. K. , Mohanty R. K.
Published: August 2005

We present a 19‐point fourth‐order finite difference method for the nonlinear second‐order system of three‐dimensional elliptic equations Auxx + Buyy + Cuzz = f, where A, B, C, are M × M diagonal matrices, on a cubic region R subject to the Dirichlet boundary conditions u(x, y, z) = u(0)(x, y, z) on ∂R. We establish, under appropriate conditions, O(h4) convergence of the difference method. Numerical examples are given to illustrate the method and its fourth‐order convergence. © 1992 John Wiley & Sons, Inc.
Explicit block iterative method for the solution of the biharmonic equation
Authors: Yousif W. S. , Evans D. J.
Published: August 2005

We develop an iterative algorithm for the solution of a finite difference approximation of the biharmonic equation over a rectangular region by using the explicit block iterative method, i.e., box over relaxation scheme. The 4‐ and 9‐point explicit blocks were considered and performance results for the two algorithms are presented.© 1993 John Wiley & Sons, Inc.
Local a posteriori error estimators for variational inequalities
Authors: Ainsworth Mark , Oden J. Tinsley , Lee C. Y.
Published: August 2005

Local a posteriori error estimators for finite element approximation of variational inequalities are derived. These are shown to provide upper bounds on the discretization error. Numerical examples are given illustrating the theoretical results. © 1993 John Wiley & Sons, Inc.
Coupling method for the exterior stationary Navier–Stokes equations
Authors: Kaitai Li , Yinnian He
Published: August 2005

We represent a new numerical method to solve the stationary Navier–Stokes equations in an unbounded domain. This technique consists in coupling the boundary integral and finite element methods. Moreover, we derive the variational formulation and well‐posedness of the coupling method and provide the convergence result for the approximate solution. © 1993 John Wiley & Sons, Inc.
Numerical method for a model of inhomogeneous muscle fibers
Authors: Milner Fabio A.
Published: August 2005

The numerical solution of a nonlinear and nonlocal system of integro‐differential equations that models the contraction of a single heterogeneous muscle fiber is studied. The method proposed is a form of the finite difference method of characteristics. Optimal error bounds are established for the resulting approximation. © 1993 John Wiley & Sons, Inc.
Picard iteration convergence analysis in a Galerkin finite element approximation of the one‐dimensional shallow water equations
Authors: Cathers B. , O'Connor B. A.
Published: August 2005

A Fourier method for analyzing iteration convergence is applied to four different formulations of the “simple” or Picard iteration procedure of a shallow water model. The model uses the Galerkin linear FE scheme in space and the implicit θ finite differencing in time. Although the numerical scheme is implicit and linearly stable for forward centering in time (θ > 0.5), it is shown that if the Picard iteration procedure is used, there may be additional operating restrictions which need to be observed to achieve iteration convergence. These restrictions effectively limit the time step (Δt) and take the form of upper bounds on θ times the Courant number (θℂ). In the 1D shallow water model, iteration convergence requires θℂ\documentclass{article}\pagestyle{empty}\begin{document}$ \le 1/ \sqrt 3 $\end{document} for two of the four iteration formulations, but there are no effective restrictions for the other two formulations. These results were confirmed by numerical experiments. There is a tradeoff between computer storage and run times. The model version with the shortest run time (due to unrestricted Δt) also requires most computer storage. © 1993 John Wiley & Sons, Inc.
A second‐order, chaos‐free, explicit method for the numerical solution of a cubic reaction problem in neurophysiology
Authors: Price W. G. , Wang Yigong , Twizell E. H.
Published: August 2005

A second order explicit method is developed for the numerical solution of the initialvalue problem w′(t) ≡ dw(t)/dt = ϕ(w), t > 0, w(0) = W0, in which the function ϕ(w) = αw(1 − w) (w − a), with α and a real parameters, is the reaction term in a mathematical model of the conduction of electrical impulses along a nerve axon. The method is based on four first‐order methods that appeared in an earlier paper by Twizell, Wang, and Price [Proc. R. Soc. (London) A 430, 541–576 (1990)]. In addition to being chaos free and of higher order, the method is seen to converge to one of the correct steady‐state solutions at w = 0 or w = 1 for any positive value of α. Convergence is monotonic or oscillatory depending on W0, α, a, and l, the parameter in the discretization of the independent variable t. The approach adopted is extended to obtain a numerical method that is second order in both space and time for solving the initial‐value boundary‐value problem ∂u/∂t = κ∂2u/∂x2 + αu(1 − u)(u − a) in which u = u(x,t). The numerical method so developed obtained the solution by solving a single linear algebraic system at each time step. © 1993 John Wiley & Sons, Inc.
Premature in‐bore separation of sabot‐jacketed projectiles
Authors: Cooke Charlie H.
Published: August 2005

The premature separation of saboted projectiles undergoing high‐mach‐number in‐bore acceleration is investigated. Results from a simplified numerical model for the problem are presented. Measured short‐term velocity history for a typical projectile is used to describe the motion of a nonuniformly accelerating piston. Two numerical algorithms for calculating the shocked flow in tront of the accelerating piston are presented. Mach number and pressure time histories for the flow afford corrective input to the sabot design process. © 1993 John Wiley & Sons, Inc.
Galerkin method for a nonlinear parabolic–elliptic system with nonlinear mixed boundary conditions
Authors: Sunmonu Adefemi
Published: August 2005

Materials which are heated by the passage of electricity are usually modeled by a nonlinear coupled system of two partial differential equations. The current equation is elliptic, while the temperature equation is parabolic. These equations are coupled one to another through the conductivities and the Joule effect. A computationally attractive discretization method is analyzed and shown to yield optimal error estimates in H1. © 1993 John Wiley & Sons, Inc.
Comparison of two cell‐centered multigrid schemes for problems with discontinuous coefficients
Authors: Ersland Brit Gunn , Teigland Rune
Published: August 2005

Two different schemes for constructing coarse‐grid operators are implemented in a linear multigrid code. In the first scheme, the construction of the coarse‐grid operators is done using a variational approach. Certain conservation properties of the fine‐grid matrices are shown to be preserved on the coarser grids by the variational construction. In the second scheme, the diffusion coefficients for the coarse grids are calculated by a simple restriction of the coefficient from the fine grid, using a flux conservation principle. The multigrid codes are then applied to solve the linear equations from an IMPES formulation of a two‐phase porous‐media flow model. A standard elliptic model problem with jump discontinuous coefficients is also solved using the two multigrid schemes. In simple cases of particular elliptic equations these two schemes are identical. However, in more general cases, such as in reservoir problems, these schemes differ. It is shown that multigrid efficiency typical of the constant coefficient cases is obtained for these problems involving discontinuous coefficients. © 1993 John Wiley & Sons, Inc.
Three‐dimensional singularities of elastic fields near vertices
Authors: Schmitz Hermann , Volk Klaus , Wendland Wolfgang
Published: August 2005

For the computation of the singular behavior of an elastic field near a three‐dimensional vertex subject to displacement boundary conditions we use a boundary integral equation of the first kind whose unknown is the boundary stress. Localization at the vertex and Mellin transformation yield a one‐dimensional integral equation on a piecewise circular curve γ in IR3 depending holomorphically on the complex Mellin parameter. The corresponding spectral points and packets of generalized eigenvectors characterize the desired stress field and are computed by a spline‐Galerkin method with graded meshes at the corner points of the curve γ. © 1993 John Wiley & Sons, Inc.
Dual finite element analysis of three‐dimensional axisymmetric elliptic problems. Part I
Authors: Hlaváček Ivan , Křížek Michal
Published: August 2005

This paper consists of two parts in which we propose two types of pure dual finite element (FE) models of a three‐dimensional (3D) axisymmetric elliptic problem with mixed boundary conditions. Using cylindrical coordinates and weighted Sobolev spaces, a dual 3D problem is transformed into a 2D problem and finite element spaces of divergence‐free vector functions are constructed with the help of a stream function. In part I of the papera priorianda posteriorierror estimates are derived for the first type of the FE model. © 1993 John Wiley & Sons, Inc.
Composite mixed finite elements on plane quadrilaterals
Authors: Bendali A. , Lahmar N.
Published: August 2005

Mixed finite elements over a plane convex quadrilateral are obtained by assembling two Raviart‐Thomas mixed finite elements over triangles. The macroelement is given by an eliminating procedure of the degrees of freedom related to the common edge to the two triangles. This procedure results in a finite element with a space of interpolating functions containing the polynomials of degree ⩽ l, where l is the greater integer for which the same property is satisfied by the relevant Raviart‐Thomas [Mathematical Aspects of Finite Element Methods, Roma 1975, I. Galligani and E. Magenes, Eds., Lecture Notes in Mathematics Vol. 606, Springer‐Verlag, Berlin, 1975] mixed finite element. The interpolation error is estimated by means of the technique of almost equivalent affine element as given by Ciavaldini and Nédélec [Rev. Fr. Autom. Inf. Recher. Opérationnelle Ser. Rouge R2, 29–45 (1974)]. © 1993 John Wiley & Sons, Inc.