bluuebunny/crossref_metadata_2025_split huggingface dataset:
Error estimates for mixed finite element methods for nonlinear parabolic problems
Authors: Chou So‐Hsiang , Li Qian 
Published: August 2005

AbstractWe 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

AbstractA 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

AbstractAn 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

AbstractWe 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

AbstractWe 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

AbstractLocal 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

AbstractWe 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

AbstractThe 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

AbstractA 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

AbstractA 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

AbstractThe 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

AbstractMaterials 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

AbstractTwo 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

AbstractFor 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

AbstractThis 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

AbstractMixed 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.
Low‐order macroelements for two‐ and three‐dimensional Stokes flow
Authors: Nafa K. , Thatcher R. W. 
Published: August 2005

AbstractThe use of low‐order elements for approximating fluid flow is attractive because all the elemental contributions can be quickly and easily obtained. One of the drawbacks is that low‐order elements often give rise to spurious pressure modes or incompatible velocity and pressure approximations. In this paper linear velocity and linear pressure elements are described for both two‐ and three‐dimensional flow that always produce stable solutions provided the elements are assembled into simple macroelements following easily used rules. Some examples of this idea are given for Stokes flow and compared with another popular low‐order method. © 1993 John Wiley & Sons, Inc.
Compact high‐order finite‐element method for elliptic transport problems with variable coefficients
Authors: MacKinnon R. J. , Langerman M. A. 
Published: August 2005

AbstractWe present a new conforming bilinear Petrov‐Galerkin finite‐element scheme for elliptic transport problems with variable coefficients. This scheme combines a generalized test function and artificial diffusion to achieve O(h4) grid‐point accuracy on uniform stencils of 3 × 3 in two dimensions without resorting to the extended stencils of high‐order elements. The method is compared with upwind and high‐order finite‐difference schemes and the standard Galerkin finite‐element method for representative test problems. © 1994 John Wiley & Sons, Inc.
Numerical study of the incremental unknowns method
Authors: Garcia Salvador 
Published: August 2005

AbstractThrough numerical experiments we explore the incremental unknowns method; linear stationary as well as evolutionary problems are considered. For linear stationary problems, the method appears as a nearly optimal two‐step preconditioning technique. At each step, we observe that the convergence behavior of the iterative methods employed is dramatically different, depending upon whether or not preconditioning is used. For linear evolutionary problems, successful and sharply accurate long‐term integration is observed when the incremental unknowns (other than that of the coarsest level) are, effectively, small quantities. Otherwise, systematical aliasing arises. © 1994 John Wiley & Sons, Inc.
Taylor‐Galerkin algorithms for viscoelastic flow: Application to a model problem
Authors: Carew E. O. , Townsend P. , Webster M. F. 
Published: August 2005

AbstractCoupled and decoupled Taylor‐Galerkin algorithms are considered for viscoelastic flow and a model problem—transient startup Poiseuille flow in a channel under a fixed pressure gradient. All algorithms reproduce the steady‐state solutions and are stable at high elasticity numbers (E). For a fixed mesh, the coupled and decoupled versions (TGC and TGD) give exceptional time‐accuracy at low elasticity numbers [to within O(1%) at E = 1] and reasonable accuracy at high elasticity numbers [to within O(10%) at E = 10, 100]. By definition, the decoupled false‐transient scheme (TGF), which uses different time scales for velocity and stress time stepping, provides a poor transient history. Where the main requirement is to compute a steady‐state algorithm efficiency is crucial. The TGF scheme attains a steady state between six to eight times faster than does the TGC scheme, and the latter is over twice as fast as the TGD form. © 1994 John Wiley & Sons, Inc.
On the relationship between the moving finite‐element procedure and best piecewise L2 fits with adjustable nodes
Authors: Baines M. J. 
Published: August 2005

AbstractIt is shown that, for a class of time‐dependent partial differential equations of the form ut = ℒ︁u, one step of the moving finite‐element (MFE) procedure corresponds to one iteration of an algorithm for obtaining best L2 fits with adjustable nodes to continuous functions. In the steady‐state limit the MFE procedure gives the best fit of ℒ︁u, with adjustable nodes, to the null function. For first‐order partial differential equations, the MFE procedure moves nodes with approximate characteristic nodal speeds. We identify an additional speed component arising directly from the L2 projection which seeks a best fit in the sense described above. © 1994 John Wiley & Sons, Inc.
Numerical inaccuracies across the interface of a nested grid
Authors: Peggion Germana 
Published: August 2005

AbstractThis article addresses and discusses the inaccuracies in finite differencing across the interface of a nested grid. Explicit schemes for the advection and diffusion equations are analyzed on the fine and coarse grids and reformulated at the interface to guarantee that the evolving solution is unaffected by the abrupt change of the spatial grid resolution. The associated errors are expressed as a function of the wavelength of the initial field distribution and the ratio between the coarse and fine grid resolution. It is found that large‐scale features of the coarse grid must supply energy to sustain the small‐scale features of the fine grid. To not deplete the large‐scale motion, a source of energy must be given at the interface in the form of a computational diffusive term with negative viscosity coefficient. On the other hand, not all the energy of the small‐scale features of the fine grid has to be transferred to the large‐scale motion, but some of it needs to be computationally dissipated at the interface. © 1994 John Wiley & Sons, Inc.
Upwind finite difference schemes for linear conservation law with memory
Authors: Lin Yanping 
Published: August 2005

AbstractIn this article we consider upwind finite difference schemes for a class of linear conservation laws with memory. Assuming the positivity of the kernel, it is proved by using the energy estimates that the upwind finite difference scheme, explicit or implicit, is stable and convergent to the real solution. The numerical results of some examples, including Burger's equation with memory, are reported; the effect of memory is also discussed based on the numerical results. © 1994 John Wiley & Sons, Inc.
Low and moderate Reynolds number transient flow simulations using space filtered Navier‐Stokes equations
Authors: Cantekin M. E. , Westerink J. J. , Luettich R. A. 
Published: August 2005

AbstractIn this study, low and moderate Reynolds number flow problems in the laminar range are solved numerically with grids that do not resolve all the significant scales of motion. Spatial averaging or filtering of the Navier‐Stokes equations and Taylor series approximations to the filtered advective terms are used in order to account for the effects of the unresolved or subgrid scales on the resolved scales. Numerical experiments with a transient 2‐D lid driven cavity flow problem, using a penalty method Galerkin finite element code, show that this approach enhances the momentum transfer properties of the numerical solution, eliminates 2Δx type oscillations, and enables the use of coarser grids. The significance and order of the terms that describe the interaction between the resolved and the subgrid scales is studied and the success of the series approximations to these terms is demonstrated. © 1994 John Wiley & Sons, Inc.
Parallel evaluation of leftmost eigenpairs of large unsymmetric matrices
Authors: Pini G. 
Published: August 2005

AbstractA parallel algorithm for the efficient calculation of m (m ⩽ 15) eigenvalues of smallest absolute magnitude for large sparse unsymmetric matrices is implemented and presented. The procedure employes a modification of the reverse simultaneous iteration scheme, which involves, among other things, the solution of m systems of linear equations. This phase is by far the most computationally demanding of the entire algorithm. However, efficient parallelization can be achieved, highly reducing the overall computational load. Numerical experiments consider the calculation of the m = 12 and m = 15 leftmost eigenvalues and eigenvectors of seven test matrices of varying size between n = 512 and n = 3564. All the computations are performed on a 4 CPU CRAY YMP8/432 machine. The accuracy of the eigenpairs found with the proposed algorithm is independent of the number of CPUs employed. Wall clock time and speed‐up measurements show that the scheme is efficient and robust and is well parallelized. In fact, average speed‐up factors of up to 3.72 were obtained. © 1994 John Wiley & Sons, Inc.
The performance of Bank‐Weiser's error estimator for quadrilateral finite elements
Authors: Ainsworth Mark 
Published: August 2005

AbstractThe error estimator proposed by Bank and Weiser is analyzed in the case of degree p finite element approximations on quadrilateral meshes. It is shown that the scheme is asymptotically exact in the energy norm for regular solutions provided that the degree of approximation is of odd order and the elements are rectangles. Perhaps surprisingly, the hypotheses are necessary, as counterexamples show. © 1994 John Wiley & Sons, Inc.
Some remarks on Zienkiewicz‐Zhu estimator
Authors: Rodríguez Rodolfo 
Published: August 2005

AbstractIn this article, Zienkiewicz‐Zhu estimator is analyzed for the piecewise linear finite element approximate solution of an elliptic problem. The estimator is proved to be equivalent to the error for the Poisson equation with a homogeneous Dirichlet boundary condition for any triangular regular mesh. No assumptions are needed about the regularity of the solution (i.e., solutions with corner singularities are not excluded). The estimator is also proved to be asymptotically exact on subdomains where the solution is smooth when parallel meshes are used. Therefore, its behavior is similar to that of other well‐known estimators. © 1994 John Wiley & Sons, Inc.
A novel numerical technique to investigate nonlinear guided waves: Approximation of nonlinear Schrödinger equation by nonperiodic pseudospectral methods
Authors: De Veronico C. , Funaro D. , Reali G. C. 
Published: August 2005

AbstractA new numerical technique for investigating light waves guided by planar nonlinear dielectric films is presented. The method implements a multidomain spectral type approximation based on orthogonal algebraic polynomials, and makes possible to deal with nonperiodic boundary conditions and discontinuities of the data, overcoming the known deficiencies of the trigonometric polynomials. Results of preliminary numerical experiments for the solution of the nonlinear Schrödinger equation are presented. © 1994 John Wiley & Sons, Inc.
An approximate factorization scheme for elliptic grid generation with control functions
Authors: Mathur J. S. , Chakrabartty S. K. 
Published: August 2005

AbstractAn Alternating Direction Implicit (ADI), Approximate Factorization (AF) scheme is presented here for the solution of the two‐dimensional elliptic partial differential equations, with control functions as source terms, used for grid generation. This scheme requires significantly less computational effort than a Successive Over Relaxation (SOR) scheme. The dependence of the choice of the acceleration parameter on the rate of convergence of the AF scheme has been studied. As an example, grids generated by this method are shown, along with a comparison of the convergence history for the present AF and SOR schemes. © 1994 John Wiley & Sons, Inc.