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We consider nondegenerate (simplicial) subdivisions (triangulation), refined near the boundary of a domain, and derive weight estimates for the Courant approximation of functions whose second order derivatives grow (near the boundary). Bibliography: 7 titles. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we have developed a fourth-order compact finite difference scheme for solving the convection–diffusion equation with Neumann boundary conditions. Firstly, we apply the compact finite difference scheme of fourth-order to discrete spatial derivatives at the interior points. Then, we present a new compact finite difference scheme for the boundary points, which is also fourth-order accurate. Finally, we use a Padé approximation method for the resulting linear system of ordinary differential equations. The presented scheme has fifth-order accuracy in the time direction and fourth-order accuracy in the space direction. It is shown through analysis that the scheme is unconditionally stable. Numerical results show that the compact finite difference scheme gives an efficient method for solving the convection–diffusion equations with Neumann boundary conditions. [Copyright &y& Elsevier]

In diffusion transport, the flux at a point is typically modeled in terms of the local gradient of a potential. When heterogeneities are present, this local model can break down and it may be more appropriate to model the diffusion flux as a weighted sum of gradients present throughout the domain. Here a discrete nonlocal flux model-consistent with control-volume implementations-is developed. This scheme is referred to as the control-volume weighted flux scheme (CVWFS). The key component is the modeling of the diffusion flux at a given control-volume face in terms of a weighted sum of gradients at that face and at faces up- and downstream. Criteria for choosing the weights are proposed. This results in numerical solution schemes in which the coefficient matrix is diagonally dominant, has positive off-diagonal elements, and zero row sums. For a particular power-law weighting scheme it is shown how the CVWFS is related to the definition of the Caputo fractional derivative and the one-shift Grunwald approximation of the Riemann-Liouville fractional derivative. On developing transients and boundary condition treatments, the accuracy and suitability of the CVWFS scheme is demonstrated by solving a number of problems governed by Caputo fractional diffusion equations. [ABSTRACT FROM AUTHOR]

A new dual-Petrov-Galerkin method is proposed, analyzed, and implemented for third and higher odd-order equations using a spectral discretization. The key idea is to use trial functions satisfying the underlying boundary conditions of the differential equations and test functions satisfying the "dual" boundary conditions. The method leads to linear systems which are sparse for problems with constant coefficients and well conditioned for problems with variable coefficients. Our theoretical analysis and numerical results indicate that the proposed method is extremely accurate and efficient and most suitable for the study of complex dynamics of higher odd-order equations. [ABSTRACT FROM AUTHOR]

This paper proposes a method of two dimenslonal analysis for the thickness-flexural vibration, where the one-dimensional finite-element method (FEM) is applied to the planar equation. This approach aims at application to the analysis of the flexural spurious vibrations of strip rectangular AT-cut plate often used in practice. The feature of the proposed method is that the two-dimensional coupled vibration can be analyzed by applying twice the one-dimensional FEM. in other words, the one-dimensional FEM is applied to the eigenmode guided by the X-edge, as well as to the resonant expression derived from the boundary condition on the Z'-edge. As a result, the considered matrix is reduced greatly in size compared to the direct analysis by the two-dimensional FEM. The result of calculation for the change of the resonant frequency due to the change of the Z'-edge ratio is compared to the result of experiment, and the usefulness of the proposed method is demonstrated. It is shown also that most of the spurious vibrations depending on the Z'-edge ratio are flexural vibrations, which is a new finding. [ABSTRACT FROM AUTHOR]