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  • 1
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    Error estimates using the cell discretization method for second-order hyperbolic equations (1997)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 13 (1997), S. 531-548 
    Details
    ISSN: 0749-159X
    Keywords: Hyperbolic second-order partial differential equations ; nonconforming finite element methods ; primal hybrid method ; cell discretization ; Mathematics and Statistics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: The cell discretization algorithm provides approximate solutions to second-order hyperbolic equations with coefficients independent of time. We obtain error estimates that show general convergence for homogeneous problems using semi-discrete approximations. A polynomial implementation of the algorithm is described that is a nonconforming extension of the finite element method that can also produce the continuous approximations of an h-p finite element method. Numerical tests are made that confirm the theoretical estimates. © 1997 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 13: 531-548, 1997
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    Analysis of finite element approximation and quadrature of Volterra integral equations (1997)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 13 (1997), S. 663-672 
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    ISSN: 0749-159X
    Keywords: finite element method ; Volterra integral equations ; Galerkin method ; projection method ; Mathematics and Statistics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: In this article we study Galerkin finite element approximations to integral equations of the Volterra type. Our prime concern is the noncoercive case, which is not covered by the standard finite element theory. The question of rates of convergence is studied for the case where an exact stiffness matrix is available, as well as the case where the latter is approximated via quadrature rules. The optimality of these rules is also considered from the point of view of the effect the choice of the quadrature has on the overall rate of convergence. © 1997 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 13: 663-672, 1997
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    Runge-Kutta characteristic methods for first-order linear hyperbolic equations (1997)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 13 (1997), S. 617-661 
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    ISSN: 0749-159X
    Keywords: characteristic methods ; Eulerian-Langrangian methods ; numerical solution of first-order hyperbolic equations ; Runge-Kutta methods ; Mathematics and Statistics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: We develop two Runge-Kutta characteristic methods for the solution of the initial-boundary value problems for first-order linear hyperbolic equations. One of the methods is based on a backtracking of the characteristics, while the other is based on forward tracking. The derived schemes naturally incorporate inflow boundary conditions into their formulations and do not need any artificial outflow boundary condition. They are fully mass conservative and can be viewed as higher-order time integration schemes improved over the ELLAM (Eulerian-Lagrangian localized adjoint method) method developed previously. Moreover, they have regularly structured, well-conditioned, symmetric, and positive-definite coefficient matrices. Extensive numerical results are presented to compare the performance of these methods with many well studied and widely used methods, including the Petrov-Galerkin methods, the streamline diffusion methods, the continuous and discontinuous Galerkin methods, the MUSCL, and the ENO schemes. The numerical experiments also verify the optimal-order convergence rates of the Runge-Kutta methods developed in this article. © 1997 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 13: 617-661, 1997
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    Parallel block iterative method for multiaquifer flow models (1997)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 13 (1997), S. 39-50 
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    ISSN: 0749-159X
    Keywords: Mathematics and Statistics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: Flow in a multiaquifer porous system can be simulated by the so-called “quasi three-dimensional” models. When heterogeneous and/or aquitards with nonlinear hydrogeologic behavior are considered, a fully numerical approach is required for the model solution. If the finite element method is used to integrate the partial differential flow equations, the final solution of large systems is required. In the present article, an original iterative solution strategy is developed. The global system is decoupled into a number of smaller subsystems consistent with the geologic structure (aquitards and aquifers) of the multiaquifer system. The aquifer and the aquitard equations are solved separately with the modified conjugate gradient and the Thomas algorithms, respectively, while the final coupled solution is obtained with an iterative procedure equivalent to a Block Jacobi scheme. The procedure can be efficiently implemented on a parallel super-computer distributing the computational load, so that two successive blocks (related to an aquifer and the underlying aquitard) are solved on the same processor. The procedure is analyzed with linear porous media, where the convergence is theoretically ensured. The results obtained with a realistic linear multiaquifer system, employing a massively parallel computer like the CRAY T3D, have pointed out the high degree of parallelization of the algorithm. Comparison with the parallel implementation of the Block SOR and Block Gauss-Seidel schemes shows that parallel Block Jacobi performs significantly better with a reduction of the elapsed times, which depends on the rate of leakage between neighboring aquifers. © 1997 John Wiley & Sons, Inc.
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    Relation between the time and space step-sizes in nonstandard finite-difference schemes for the Fisher equation (1997)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 13 (1997), S. 51-55 
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    ISSN: 0749-159X
    Keywords: Mathematics and Statistics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: A positivity condition is used to obtain functional relations between the time and space step-sizes for nonstandard finite-difference models of the Fisher partial differential equation. An upper bound is also derived for the solutions to the difference equations. © 1997 John Wiley & Sons, Inc.
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    Accelerated multigrid high accuracy solution of the convection-diffusion equation with high Reynolds number (1997)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 13 (1997), S. 77-92 
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    ISSN: 0749-159X
    Keywords: Mathematics and Statistics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: A fourth-order compact finite difference scheme is employed with the multigrid algorithm to obtain highly accurate numerical solution of the convection-diffusion equation with very high Reynolds number and variable coefficients. The multigrid solution process is accelerated by a minimal residual smoothing (MRS) technique. Numerical experiments are employed to show that the proposed multigrid solver is stable and yields accurate solution for high Reynolds number problems. We also show that the MRS acceleration procedure is efficient and the acceleration cost is negligible. © 1997 John Wiley & Sons, Inc.
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    The study on the nonlinear computations of the DQ and DC methods (1997)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 13 (1997), S. 57-75 
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    ISSN: 0749-159X
    Keywords: Mathematics and Statistics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: This article points out that the differential quadrature (DQ) and differential cubature (DC) methods, due to their global domain property, are more efficient for nonlinear problems than the traditional numerical techniques such as finite element and finite difference methods. By introducing the Hadamard product of matrices, we obtain an explicit matrix formulation for the DQ and DC solutions of nonlinear differential and integro-differential equations. Due to its simplicity and flexibility, the present Hadamard product approach makes the DQ and DC methods much easier to use. Many studies on the Hadamard product can be fully exploited for the DQ and DC nonlinear computations. Furthermore, we first present the SJT product of matrix and vector to compute accurately and efficiently the Frechet derivative matrix in the Newton-Raphson method for the solution of the nonlinear formulations. We also propose a simple approach to simplify the DQ or DC formulations for some nonlinear differential operators and, thus, the computational efficiency of these methods is significantly improved. We give the matrix multiplication formulas to efficiently compute the weighting coefficient matrices of the DC method. The spherical harmonies are suggested as the test functions in the DC method to handle the nonlinear differential equations occurring in global and hemispheric weather forecasting problems. Some examples are analyzed to demonstrate the simplicity and efficiency of the presented techniques. It is emphasized that the innovations presented are applicable to the nonlinear computations of the other numerical methods as well. © 1997 John Wiley & Sons, Inc.
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    A residual-based a posteriori error estimator for the Ciarlet-Raviart formulation of the first biharmonic problem (1997)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 13 (1997), S. 93-111 
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    ISSN: 0749-159X
    Keywords: Mathematics and Statistics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: In this article we propose a residual-based estimator for the psi-omega formulation of the biharmonic problem. We show how an appropriate modification of Verfürth methodology gives the proper scaling of the residuals leading to both lower and upper estimation. Numerical examples confirm the viability of the method. © 1997 John Wiley & Sons, Inc.
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  • 9
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    An improved coarse-mesh nodal integral method for partial differential equations (1997)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 13 (1997), S. 113-145 
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    ISSN: 0749-159X
    Keywords: coarse-mesh ; nonlinear PDEs ; Burgers equation ; inherent upwinding ; Mathematics and Statistics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: An improved variation of the nodal integral method to solve partial differential equations has been developed and implemented. Rather than treating all of the nonlinear terms as the so-called pseudo-source terms (to be approximated), in this modified version of the nodal integral method, by approximating part of the nonlinear terms in terms of the discrete variable(s) that ultimately result at the end of the formulation process, some or all of the nonlinear terms are kept on the left-hand side in the transverse-integrated equations, which are to be solved analytically. Application of the method to solve the Burgers equation leads to exponential variation within the nodes and shows that the resulting scheme has inherent upwinding. Reconstruction of node interior solution - as a function of one independent variable, and averaged in all others - makes it possible to obtain rather accurate solutions even on a fine scale. Results of the numerical analysis and comparison with results of other methods reported in the literature show that the new method is comparable and sometimes better in accuracy than the currently used schemes. Extension to multidimensional, time-dependent problems is straightforward. © 1997 John Wiley & Sons, Inc.
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    Analysis and convergence of a MAC-like scheme for the generalized Stokes problem (1997)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 13 (1997), S. 147-162 
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    ISSN: 0749-159X
    Keywords: covolume methods ; MAC schemes ; mixed methods ; Stokes problem ; Mathematics and Statistics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: We introduce a MAC-like scheme (a covolume method on rectangular grids) for approximating the generalized Stokes problem on an axiparallel domain. Two staggered grids are used in the derivation of the discretization. The velocity is approximated by conforming bilinears over rectangular elements, and the pressure by piecewise constants over macro-rectangular elements. The error in the velocity in the H1 norm and the pressure in the L2 norm are shown to be of first order, provided that the exact velocity is in H2 and the exact pressure in H1, and that the partition family of the domain is regular. © 1997 John Wiley & Sons, Inc.
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    A least-square mixed method for Stokes equations (1997)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 13 (1997), S. 191-199 
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    ISSN: 0749-159X
    Keywords: Mathematics and Statistics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: We prove the convergence of a least-square mixed method for Stokes equations by use of an operator theoretic approach. The method does not require LBB condition on the finite dimensional subspaces. The resulting bilinear form is symmetric and positive definite, which leads to optimal convergence and the h-2 condition number. © 1997 John Wiley & Sons, Inc.
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    On the convergence of a combined finite volume-finite element method for nonlinear convection-diffusion problems (1997)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 13 (1997), S. 163-190 
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    ISSN: 0749-159X
    Keywords: Mathematics and Statistics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: We present the convergence analysis of an efficient numerical method for the solution of an initial-boundary value problem for a scalar nonlinear conservation law equation with a diffusion term. Nonlinear convective terms are approximated with the aid of a monotone finite volume scheme considered over the finite volume mesh dual to a triangular grid, whereas the diffusion term is discretized by piecewise linear conforming triangular elements. Under the assumption that the triangulations are of weakly acute type, with the aid of the discrete maximum principle, a priori estimates, and some compactness arguments based on the use of the Fourier transform with respect to time, the convergence of the approximate solutions to the exact solution is proved, provided that the mesh size tends to zero. © 1997 John Wiley & Sons, Inc.
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    How to generate local refinements of unstructured tetrahedral meshes satisfying a regularity ball condition (1997)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 13 (1997), S. 201-214 
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    ISSN: 0749-159X
    Keywords: tetrahedral meshes ; local refinements ; finite elements ; Mathematics and Statistics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: A simple algorithm that generates local refinements of tetrahedral meshes is proposed. Refinements are made by tetrahedra, which are subdivided into eight, four, three, or two smaller tetrahedra. We prove that a regularity ball condition is satisfied for the refined mesh. This guarantees that tetrahedra do not become flat when the mesh size tends to zero. © 1997 John Wiley & Sons, Inc.
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    Numerical solution of the generalized Stokes problem for the flow in a channel (1997)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 13 (1997), S. 237-256 
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    ISSN: 0749-159X
    Keywords: flow in a channel ; separation of variables ; incremental unknowns ; Mathematics and Statistics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: In this article we introduce the separation of variables in the two-dimensional generalized Stokes problem. -νΔu + αu + inverted delta p = f, for the flow in a channel. Also for the first time, we discuss the implementation of the Incremental Unknowns Method with a data structure of Compressed Column Storage. Two examples of application of the Incremental Unknowns method for this problem are presented in which we compare the CPU times of three methods: Conjugate Gradient (CG), Incremental Unknowns (IU), and Uzawa Algorithm (Uzawa). © 1997 John Wiley & Sons, Inc.
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    Mixed finite volume methods for semiconductor device simulation (1997)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 13 (1997), S. 215-236 
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    ISSN: 0749-159X
    Keywords: Mathematics and Statistics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: We deal with the two-dimensional numerical solution of the Van Roosbroeck system, widely employed in modern semiconductor device simulation. Using the well-known Gummel's decoupled algorithm leads to the iterative solution of a nonlinear Poisson equation for the electric potential and two linearized continuity equations for the electron and hole current densities. The numerical approximation is based on the dual mixed formulation for a self-adjoint second-order elliptic operator by using the Raviart-Thomas (RT) finite elements of lowest degree on a triangular partition of the device domain. In this article, we propose a suitable variant of the RT method, based on the diagonalization of the element mass matrix. This is achieved by use of an appropriate numerical integration that eliminates the fluxes and gives rise to a cell-centered finite volume scheme for the scalar unknown with the same approximation properties of the mixed approach, but at a reduced computational cost. The above procedure suggests also a natural way to introduce in the frame of the classical Box Method (BM) suitable vector basis functions (edge elements) to represent the current field over each mesh triangle. This issue may be profitably employed both as a postprocessing tool, as well as a technique for solving the current continuity equations when source terms depending on the current itself are included in the mathematical model. Simulations of realistic semiconductor devices are then included to demonstrate the accuracy and stability of the new method. © 1997 John Wiley & Sons, Inc.
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    A frequency accurate spatial derivative finite difference approximation (1997)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 13 (1997), S. 549-560 
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    ISSN: 0749-159X
    Keywords: frequency accurate methods ; finite difference approximations ; fourier transforms ; partial differential equations ; Mathematics and Statistics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: We present a method for designing spatial derivative approximations that achieves a priori accuracy in the spatial frequency domain. We use a general, average value approximation with undetermined coefficients together with a set of constraints that ensure convergence and consistency to formulate a constrained optimal fitting problem. These constraints lead to a linear matrix formulation. We apply the method to the design of spatial approximations for simulating equations with wavelike solutions using both an explicit central difference approximation (which has no phase error) and an upwind design where the level of dissipation can be specified by the designer. © 1997 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 13: 549-560, 1997
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    On the Eulerian-Lagrangian formulation of balance equations in porous media (1997)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 13 (1997), S. 505-530 
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    ISSN: 0749-159X
    Keywords: Porous medium ; modeling ; transport phenomena ; Eulerian-Lagrangian formulation ; primary variables ; reduced velocities ; Mathematics and Statistics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: The article presents a general approach to modeling the transport of extensive quantities in the case of flow of multiple multicomponent fluid phases in a deformable porous medium domain under nonisothermal conditions. The models are written in a modified Eulerian-Lagrangian formulation. In this modified formulation, the material derivatives are written in terms of modified velocities. These are the velocities at which the various phase and component variables propagate in the domain, along their respective characteristic curves. It is shown that these velocities depend on the heterogeneity of various solid matrix and fluid properties. The advantage of this formulation, with respect to the usually employed Eulerian one, is that numerical dispersion, associated with the advective fluxes of extensive quantities, are eliminated. The methodology presented in the article shows how the Eulerian-Lagrangian formulation is written in terms of the relatively small number of primary variables of a transport problem. © 1997 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 13: 505-530, 1997
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    Front tracking for two-phase flow in reservoir simulation by adaptive mesh (1997)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 13 (1997), S. 673-697 
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    ISSN: 0749-159X
    Keywords: porous media ; adaptive mesh ; MUSCL scheme ; OOP ; Mathematics and Statistics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: In this article, an algorithm for the numerical approximation of two-phase flow in porous media by adaptive mesh is presented. A convergent and conservative finite volume scheme for an elliptic equation is proposed, together with the finite difference schemes, upwind and MUSCL, for a hyperbolic equation on grids with local refinement. Hence, an IMPES method is applied in an adaptive composite grid to track the front of a moving solution. An object-oriented programmation technique is used. The computational results for different examples illustrate the efficiency of the proposed algorithm. © 1997 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 13: 673-697, 1997
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    A parallel spectral Fourier-Nonlinear Galerkin algorithm for simulation of turbulence (1997)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 13 (1997), S. 699-715 
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    ISSN: 0749-159X
    Keywords: parallel algorithm ; domain decomposition ; Local Fourier Basis ; Nonlinear Galerkin Method ; Mathematics and Statistics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: We present a high-order parallel algorithm, which requires only the minimum interprocessor communication dictated by the physical nature of the problem at hand. The parallelization is achieved by domain decomposition. The discretization in space is performed using the Local Fourier Basis method. The continuity conditions on the interfaces are enforced by adding homogeneous solutions. Such solutions often have fast decay properties, which can be utilized to minimize interprocessor communication. In effect, the predominant part of the computation is performed independently in the subdomains (processors) or using only local communication. A novel element of the present parallel algorithm is the incorporation of a Nonlinear Galerkin strategy to accelerate the computation and stabilize the time integration process. The basic idea of this approach consists of decomposition of the variables into large scale and small scale components with different treatment of these large and small scales. The combination of the Multidomain Fourier techniques with the Nonlinear Galerkin (NLG) algorithm is applied here to solve incompressible Navier-Stokes equations. Results are presented on direct numerical simulation of two-dimensional homogeneous turbulence using the NLG method. © 1997 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 13: 699-715, 1997
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    Analysis of strain-pressure finite element methods for the Stokes problem (1997)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 13 (1997), S. 717-730 
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    ISSN: 0749-159X
    Keywords: Stokes ; enhanced strains ; inf-sup condition ; Mathematics and Statistics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: An analysis of some nonconforming approximations of the Stokes problem is presented. The approximations are based on a strain-pressure variational formulation. In particular, a convergence and stability result for a method recently proposed by Bathe and Pantuso is provided. © 1997 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 13: 717-730, 1997
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    A consistent approach to variable bubble-point systems (1997)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 13 (1997), S. 1-18 
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    ISSN: 0749-159X
    Keywords: Mathematics and Statistics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: Here, it is shown that the “traditional approach” to variable bubble-point problems, using black-oil models, is not consistent, because it violates the “bubble-point conservation law.” In order to have a consistent approach, it is necessary to incorporate shocks - discussed in previous papers - in which the bubble-point is discontinuous. A “consistent approach” is applied to specific examples, and results compared with those of the “traditional” one. The conclusion that the “traditional approach” generally yields large errors for the production rates and other parameters of interest in the oil industry, is reached. © 1997 John Wiley & Sons, Inc.
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    Vorticity-velocity formulation for the Navier-Stokes equations: The three-dimensional case (1997)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 13 (1997), S. 19-38 
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    ISSN: 0749-159X
    Keywords: Mathematics and Statistics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: In this article, we propose a mixed method for the vorticity-velocity formulation of the stationary Stokes and Navier-Stokes equations in space dimension three, the unknowns being the vorticity and the velocity of the fluid. We give a similar variational formulation for the nonstationary Stokes equations in the vorticity-velocity variables. © 1997 John Wiley & Sons, Inc.
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    An iterative penalty method for the least squares solution of boundary value problems (1997)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 13 (1997), S. 257-281 
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    ISSN: 0749-159X
    Keywords: boundary value problems ; collocation least squares method ; augmented Lagrangian method ; Uzawa's algorithm ; preconditioned conjugate gradient ; Mathematics and Statistics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: This article is concerned with iterative techniques for linear systems of equations arising from a least squares formulation of boundary value problems. In its classical form, the solution of the least squares method is obtained by solving the traditional normal equation. However, for nonsmooth boundary conditions or in the case of refinement at a selected set of interior points, the matrix associated with the normal equation tends to be ill-conditioned. In this case, the least squares method may be formulated as a Powell multiplier method and the equations solved iteratively. Therein we use and compare two different iterative algorithms. The first algorithm is the preconditioned conjugate gradient method applied to the normal equation, while the second is a new algorithm based on the Powell method and formulated on the stabilized dual problem. The two algorithms are first compared on a one-dimensional problem with poorly conditioned matrices. Results show that, for such problems, the new algorithm gives more accurate results. The new algorithm is then applied to a two-dimensional steady state diffusion problem and a boundary layer problem. A comparison between the least squares method of Bramble and Schatz and the new algorithm demonstrates the ability of the new method to give highly accurate results on the boundary, or at a set of given interior collocation points without the deterioration of the condition number of the matrix. Conditions for convergence of the proposed algorithm are discussed. © 1997 John Wiley & Sons, Inc.
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    Coiflet interpolation and approximate solutions of elliptic partial differential equations (1997)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 13 (1997), S. 303-320 
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    ISSN: 0749-159X
    Keywords: Mathematics and Statistics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: In this article, we prove a higher order interpolation result for square-integrable functions by using generalized coiflets. Convergence of approximation by using generalized coiflets is shown. Applications to wavelet-Galerkin approximation of elliptic partial differential equations and some numerical examples are also given. © 1997 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 13:303-320, 1997.
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    A preconditioning strategy for boundary element Galerkin methods (1997)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 13 (1997), S. 283-301 
    Details
    ISSN: 0749-159X
    Keywords: boundary element method ; preconditioner ; conjugate gradient method ; mesh grading ; Mathematics and Statistics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: The Dirichlet and Neumann problems for the Laplacian are reformulated in the usual way as boundary integral equations of the first kind with symmetric kernels. These integral equations are solved using Galerkin's method with piecewise-constant and piecewise-linear boundary elements, respectively. In both cases, the stiffness matrix is symmetric and positive-definite, and has a condition number of order N, the number of degrees of freedom. By contrast, the condition number of the product of the two stiffness matrices is bounded independently of N. Hence, we can use the Neumann stiffness matrix to precondition the Dirichlet stiffness matrix, and vice versa. © 1997 John Wiley & Sons, Inc.
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    On preconditioned Krylov subspace methods for discrete convection-diffusion problems (1997)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 13 (1997), S. 321-330 
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    ISSN: 0749-159X
    Keywords: convection-diffusion equation ; preconditioners ; Krylov subspace methods ; Mathematics and Statistics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: We study nonstationary iterative methods for solving preconditioned systems arising from discretizations of the convection-diffusion equation. The preconditioners arise from Gauss-Seidel methods applied to the original system. It is shown that the performance of the iterative solvers is affected by the relationship of the ordering of the underlying grid and the direction of the fow associated with the differential operator. Specifically, only those orderings that follow the fow give fast iterative solvers. © 1997 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 13:321-330
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    A Gauss-Galerkin finite-difference method for singular partial differential equations in two space variables (1997)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 13 (1997), S. 331-355 
    Details
    ISSN: 0749-159X
    Keywords: Gauss-Galerkin ; finite-difference ; semi-discrete difference approximation ; weak convergence ; Mathematics and Statistics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: A Gauss-Galerkin finite-difference method is proposed for the numerical solution of a class of linear, singular parabolic partial differential equations in two space dimensions. The method generalizes a Gauss-Galerkin method previously used for treating similar singular parabolic partial differential equations in one space dimension. Two test problems are studied and the numerical results are presented. These numerical results are encouraging and suggest that the proposed method is efficient in treating singular parabolic partial differential equations of the type considered here. © 1997 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 13: 331-355, 1997
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    A family of fourth-order parallel splitting methods for parabolic partial differential equations (1997)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 13 (1997), S. 357-373 
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    ISSN: 0749-159X
    Keywords: heat equation ; finite differences ; parallel splitting ; fourth-order numerical methods ; avoiding complex arithmetic ; Mathematics and Statistics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: A family of numerical methods which are L-stable, fourth-order accurate in space and time, and do not require the use of complex arithmetic is developed for solving second-order linear parabolic partial differential equations. In these methods, second-order spatial dderivatives are approximated by fourth-order finite-difference approximations, and the matrix exponential function is approximated by a rational approximation consisting of three parameters. Parallel algorithms are developed and tested on the one-dimensional head equation, with constant coefficients, subject to homogeneous and time-dependent boundary conditions. These methods are also extended to two- and three-dimensional heat equations, with constant coefficients, subject to homogeneous boundary conditions. © 1997 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 13: 357-373, 1997
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    Galerkin method for a Stefan-type problem in one space dimension (1997)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 13 (1997), S. 393-416 
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    ISSN: 0749-159X
    Keywords: Galerkin method ; Stefan problem ; fixed domain method ; optimal error estimates ; Mathematics and Statistics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: Based on a Landau-type transformation, both continuous and discrete in time L2-Galerkin methods are applied to a single-phase Stefan-type problem in one space dimension. Optimal rates of convergence in Lℵ, L∞, and H1-norms are derived and computational results are presented. © 1997 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 13: 393-416, 1997
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    A highly accurate numerical solution of a biharmonic equation (1997)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 13 (1997), S. 375-391 
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    ISSN: 0749-159X
    Keywords: Biharmonic equation ; finite difference ; high-order accuracy ; Mathematics and Statistics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: The coefficients for a nine-point high-order accuracy discretization scheme for a biharmonic equation ∇ 4u = f(x, y) (∇2 is the two-dimensional Laplacian operator) are derived. The biharmonic problem is defined on a rectangular domain with two types of boundary conditions: (1) u and ∂2u/∂n2 or (2) u and ∂u/part;n (where ∂/part;n is the normal to the boundary derivative) are specified at the boundary. For both considered cases, the truncation error for the suggested scheme is of the sixth-order O(h6) on a square mesh (hx = hy = h) and of the fourth-order O(h4xh2xh2y h4y) on an unequally spaced mesh. The biharmonic equation describes the deflection of loaded plates. The advantage of the suggested scheme is demonstrated for solving problems of the deflection of rectangular plates for cases of different boundary conditions: (1) a simply supported plate and (2) a plate with built-in edges. In order to demonstrate the high-order accuracy of the method, the numerical results are compared with exact solutions. © John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 13: 375-391, 1997
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    A legendre pseudospectral method for a PDE model for the dynamics of infectious diseases (1997)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 13 (1997), S. 417-432 
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    ISSN: 0749-159X
    Keywords: Legendre pseudospectral method ; infectious diseases ; Mathematics and Statistics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: A Legendre pseudospectral method is developed for a PDE model for the dynamics of infectious diseases. The stability and the convergence rate of the method are studied. © 1997 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 13: 417-432, 1997
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    Convergence analysis of a staggered pressure correction scheme for viscous incompressible flows (1997)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 13 (1997), S. 459-482 
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    ISSN: 0749-159X
    Keywords: Convergence analysis ; pressure correction scheme ; viscous incompressible flows ; Mathematics and Statistics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: This article performs the convergence analysis of a staggered pressure correction scheme for solving unsteady viscous incompressible flows in a bounded domain using the energy method. Error estimates for the numerical velocity field of the difference scheme are established. The analysis adopts the method of bounded truncation functions as advocated by Arnold et al. © 1997 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 13: 459-482, 1997
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    On the convergence of a certain type of nonlinear lumped energy equations (1997)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 13 (1997), S. 433-443 
    Details
    ISSN: 0749-159X
    Keywords: Mathematics and Statistics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: Recently, Campo and Lacoa [Campo & Lacoa, Numerical Methods Partial Differential Eq., 11, 275 (1995)] proposed a simple iterative procedure to solve approximately a forced convection heat-transfer problem inside a tube subjected to nonlinear convective boundary conditions. Their technique relied on solutions of a one-dimensional ordinary differential equation of first order to estimate the behavior of the solution of the two-dimensional parabolic partial differential equation. The recursive steps, proposed by Campo and Lacoa, can be combined in a single fixed-point iteration formula thus facilitating the study of its properties. In this note, we present a short analysis of the convergence of the Campo-Lacoa equation and give alternatives to guarantee and improve the convergence patterns. Our results show that the Picard's iterative method converges for all values of Z in the region of thermal development, e.g., 0 ≤ Z ≤ 1; however, the convergence rate tends to diminish as Z increases. To guarantee convergence for larger values of Z, a damped-Picard's iteration may be adopted. Moreover, to increase the rate of convergence, a Newton's iteration is proposed. A detailed comparison in terms of accuracy and CPU time is provided. © 1997 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 13: 433-443, 1997
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    A nonconforming mixed finite element for second-order elliptic problems (1997)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 13 (1997), S. 445-457 
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    ISSN: 0749-159X
    Keywords: nonconforming mixed finite elements ; nonconforming piecewise quadratic approximations ; second-order elliptic problems ; Mathematics and Statistics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: In a recent work, Hiptmair [Mathematisches Institut, M9404, 1994] has constructed and analyzed a family of nonconforming mixed finite elements for second-order elliptic problems. However, his analysis does not work on the lowest order elements. In this article, we show that it is possible to construct a nonconforming mixed finite element for the lowest order case. We prove the convergence and give estimates of optimal order for this finite element. Our proof is based on the use of the properties of the so-called nonconforming bubble function to control the consistency terms introduced by the nonconforming approximation. We further establish an equivalence between this mixed finite element and the nonconforming piecewise quadratic finite element of Fortin and Soulie [J. Numer. Methods Eng., 19, 505-520, 1983]. © 1997 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 13: 445-457, 1997
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    Analysis of expanded mixed methods for fourth-order elliptic problems (1997)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 13 (1997), S. 483-503 
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    ISSN: 0749-159X
    Keywords: Expanded mixed methods ; finite elements ; error estimates ; fourth-order elliptic problems ; Mathematics and Statistics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: The recently proposed expanded mixed formulation for numerical solution of second-order elliptic problems is here extended to fourth-order elliptic problems. This expanded formulation for the differential problems under consideration differs from the classical formulation in that three variables are treated, i.e., the displacement, the stress, and the moment tensors. It works for the case where the coefficient of the differential equations is small and does not need to be inverted, or for the case in which the stress tensor of the equations does not need to be symmetric. Based on this new formulation, various mixed finite elements for fourth-order problems are considered; error estimates of quasi-optimal or optimal order depending upon the mixed elements are derived. Implementation techniques for solving the linear system arising from these expanded mixed methods are discussed, and numerical results are presented. © 1997 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 13: 483-503, 1997
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    Approximation of Navier-Stokes incompressible flow using a spectral element method with a local discretization in spectral space (1997)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 13 (1997), S. 587-599 
    Details
    ISSN: 0749-159X
    Keywords: Domain decomposition ; spectral element ; incompressible Navier Stokes ; legendre polynomials ; Mathematics and Statistics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: A spectral element technique is examined, which builds upon a local discretization within the spectral space. To approximate a given system of equations the domain is subdivided into nonoverlapping quadrilateral elements, and within each element a discretization is found in the spectral space. The difference is that the test functions are divided into the higher-order polynomials, which have zero boundaries and lower-order polynomials, which are nonzero on one boundary. The method is examined for Navier-Stokes incompressible flow for fluid flow within a driven cavity and for flow over a backstep. © 1997 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 13: 587-599, 1997
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    A posteriori error estimators for the steady incompressible Navier-Stokes equations (1997)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 13 (1997), S. 561-574 
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    ISSN: 0749-159X
    Keywords: A posteriori error estimators ; mixed finite elements methods ; Navier-Stokes equations ; automatic mesh refinement ; Mathematics and Statistics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: A residual-based a posteriori error estimator for finite element discretizations of the steady incompressible Navier-Stokes equations in the primitive variable formulation is discussed. Though the estimator is similar to existing ones, an alternate derivation is presented, involving an abstract estimate that may prove of some intrinsic value. The estimator is particularized to Hood-Taylor and modified Hood-Taylor finite elements and proved to be a global upper bound (up to a positive multiplicative constant) of the true error. Numerical examples are provided to illustrate the performance of the resulting mesh adaptation process. © 1997 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 13: 561-574, 1997
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    Optimal rectangular MITC finite elements for Reissner-Mindlin plates (1997)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 13 (1997), S. 575-585 
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    ISSN: 0749-159X
    Keywords: Finite elements ; plates 65N30 ; 73K10 ; Mathematics and Statistics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: In recent years a family of finite elements named mixed interpolated tensorial components (MITC) has been introduced for the numerical approximation of Reissner-Mindlin plates. The elements have been proved to be locking free. In this article, we consider the MITC rectangular finite elements and show that it is possible to reduce the number of internal degrees of freedom in the approximation of the rotation field without losing order of convergence. Our mathematical analysis is carried out combining some results for the Stokes problem with the special features of the MITC finite elements. © 1997 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 13: 575-585, 1997
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    Application of circuit analysis programs to systems of partial differential equations (1997)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 13 (1997), S. 601-615 
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    ISSN: 0749-159X
    Keywords: network analog ; partial differential equation ; circuit analysis ; SPICE ; Burger's equation ; Mathematics and Statistics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: A general approach for solving systems of time domain partial differential equations using circuit analysis programs is described. The approach is then used to solve a nonlinear one-dimensional transient fluid flow problem. Using the general purpose circuit analysis program SPICE, the approach is fully implicit and should provide a convenient method for physical simulations in one dimension. © 1997 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 13: 601-615, 1997
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