Finite volume methods with local refinement for convection-diffusion problems

Computing - Based on approximation of the balance relation for convection-diffusion problems, finite difference schemes on rectangular locally refined grids are derived and studied. A priori...     

Isoparametric finite difference energy method for plate bending problems

This paper incorporates the concept of isoparametry in finite difference energy method making it more powerful and versatile to tackle complex plate bending problems with curved boundaries. This approach overcomes the drawbacks of the finite difference energy method and its application to practical problems is now feasible. In order to estimate the accuracy and reliability of the present formulation several isotropic plates with a variety of planform are solved and the results are compared with the existing analytical and numerical solutions.     

A finite difference discretization method for elliptic problems on composite grids

In this paper we discusss a simple finite difference method for the discretization of elliptic boundary value problems on composite grids. For the model problem of the Poisson equation we prove stability of the discrete operator and bounds for the global discretization error. These bounds clearly show how the discretization error depends on the grid size of the coarse grid, on the grid size of the local fine grid and on the order of the interpolation used on the interface. Furthermore, the constants in these bounds do not depend on the quotient of coarse grid size and fine grid size. We also discuss an efficient solution method for the resulting composite grid algebraic problem.     

Adaptive finite element heterogeneous multiscale method for homogenization problems

In this paper we present an a posteriori error analysis for elliptic homogenization problems discretized by the finite element heterogeneous multiscale method. Unlike standard finite element methods, our discretization scheme relies on macro- and microfinite elements. The desired macroscopic solution is obtained by a suitable averaging procedure based on microscopic data. As the macroscopic data (such as the macroscopic diffusion tensor) are not available beforehand, appropriate error indicators have to be defined for designing adaptive methods. We show that such indicators based only on the available macro- and microsolutions (used to compute the actual macrosolution) can be defined, allowing for a macroscopic mesh refinement strategy which is both reliable and efficient. The corresponding a posteriori estimates for the upper and lower bound are derived in the energy norm. In the case of a uniformly oscillating tensor, we recover the standard residual-based a posteriori error estimate for the finite element method applied to the homogenized problem. Numerical experiments confirm the efficiency and reliability of the adaptive multiscale method.     

Exponential difference schemes for solving boundary-value problems for convection-diffusion type equations

The paper considers the numerical solution of boundary-value problems for multidimensional convection-diffusion type equations (CDEs). Such equations are useful for various physical processes in solids, liquids and gases. A new approach to the spatial approximation for such equations is proposed. This approach is based on an integral transformation of second-order one-dimensional differential operators. A linear version of CDE was chosen for simplicity of the analysis. In this setting, exponential difference schemes were constructed, algorithms for their implementation were developed, a brief analysis of the stability and convergence was made. This approach was numerically tested for a two-dimensional problem of motion of metallic particles in water flow subject to a constant magnetic field.     

A discontinuous finite difference streamline diffusion method for time-dependent hyperbolic problems

In this article, a new finite element method, discontinuous finite difference streamline diffusion method (DFDSD), is constructed and studied for first-order linear hyperbolic problems. This method combines the benefit of the discontinuous Galerkin method and the streamline diffusion finite element method. Two fully discrete DFDSD schemes (Euler DFDSD and Crank–Nicolson (CN) DFDSD) are constructed by making use of the difference discrete method for time variables and the discontinuous streamline diffusion method for space variables. The stability and optimal L² norm error estimates are established for the constructed schemes. This method makes contributions to the discontinuous methods. Finally, a numerical example is provided to show the benefit of high efficiency and simple implementation of the schemes.     

A finite difference method for a Stefan problem

Many physical problems involve initial boundary value problems for parabolic differential equations in which part of the boundary is not given a priori but is found as part of the solution. These problems have been considered under the name of «Stefan problems». Stefan problems occur in such physical processes as the melting of solids and the crystallizing of liquids. Existence and uniqueness for various one dimensional Stefan problems have been shown by several authors ([1], [4], [5], [8], [9]). Some multidimensional Stefan problems were considered by ([6], [11]). The purpose of this paper is to approximate the solution of a Stefan problem using an implicit finite difference analog for the heat equation and an explicit finite difference analog for the differential equation describing the free boundary. Also, we shall show that the finite difference solution we obtain converges uniformly to the actual solution. Numerous numerical schemes for various Stefan problems have been successfully employed by several authors ([2], [3], [10], [12], [14], [16], [17]). In Chapter I we formulate a continuous time Stefan problem, and in Chapter II we describe a finite difference scheme for approximating results. Chapter IV contains the main result which shows that the finite difference solution converges to the actual solution. Finally, in Chapter V we give a numerical example.     

Avoiding slave points in an adaptive refinement procedure for convection-diffusion problems in 2D

A finite difference method is presented for singularly perturbed convection-diffusion problems with discretization error estimate of nearly second order. In a standard patched adaptive refinement method certain slave nodes appear where the approximation is done by interpolating the values of the approximate solution at adjacent nodes. This deteriorates the accuracy of truncation error. In order to avoid the slave points we change the stencil at the interface points from a cross to a skew one. The efficiency of this technique is illustrated by numerical experiments in 2D.     

Adaptive mesh refinement in strain localization problems

An adaptive meshing method tailored to problems of strain localization is given. The adaption strategy consists of equi-distributing the variation of the velocity field over the elements of the mesh. A heuristic justification for the use of variations as indicators is advanced, and possible connections with interpolation error bounds are discussed. Meshes are constructed by Delaunay triangulation. It is shown how the Hu-Washizu principle determines a consistent transfer operator for the state variables. Examples of application are given which demonstrate the versatility of the method.     

The application of a general finite difference method to some boundary value problems

This paper presents in full a method, based on Taylor's expansion, of applying finite difference techniques to boundary value problems of complex geometry. It demonstrates both the ease with which boundary conditions are incorporated and how error analyses may be performed. A range of second and fourth order examples are presented to illustrate the versatility of the method.     

Adaptive mesh refinement using piecewise-linear shape functions based on the blending function method

Use of quadrilateral elements for finite element mesh refinement can lead either to so-called irregular meshes or the necessity of adjustments between finer and coarser parts of the mesh necessary. In the case of irregular meshes, constraints have to be introduced in order to maintain continuity of the displacements. Introduction of finite elements based on blending function interpolation shape functions using piecewise boundary interpolation avoids these problems. This paper introduces an adaptive refinement procedure for these types of elements. The refinement is anh-method. Error estimation is performed using the Zienkiewicz-Zhu method. The refinement is controlled by a switching function representation. The method is applied to the plane stress problem. Numerical examples are given to show the efficiency of the methodology.     

Transient flow control for an artificial open channel based on finite difference method

The particular challenges of modeling control systems for the middle route of the south-to-north water transfer project are illustrated.Open channel dynamics are approximated by well-known Saint-Venant nonlinear partial differential equations.For better control purpose,the finite difference method is used to discretize the Saint-Venant equations to form the state space model of channel system.To avoid calculation divergence and improve control stability,balanced model reduction together with poles placement...     

Convergence of the mimetic finite difference method for eigenvalue problems in mixed form

Optimal convergence rates for the mimetic finite difference method applied to eigenvalue problems in mixed form are proved. The analysis is based on a new a priori error bound for the source problem and relies on the existence of an appropriate elemental lifting of the mimetic discrete solution. Compared to the original convergence analysis of the method, the new a priori estimate does not require any extra regularity assumption on the right-hand side of the source problem. Numerical results confirming the optimal behavior of the method are presented.     

The adaptive finite element method based on multi-scale discretizations for eigenvalue problems

In this paper, adaptive finite element methods for differential operator eigenvalue problems are discussed. For multi-scale discretization schemes based on Rayleigh quotient iteration (see Scheme 3 in [Y. Yang, H. Bi, A two-grid discretization scheme based on shifted-inverse power method, SIAM J. Numer. Anal. 49 (2011) 1602–1624]), a reliable and efficient a posteriori error indicator is given, in addition, a new adaptive algorithm based on the multi-scale discretizations is proposed, and we apply the algorithm to the Schrödinger equation for hydrogen atoms. The algorithm is performed under the package of Chen, and satisfactory numerical results are obtained.     

An explicit finite element method for convection-diffusion equations using rational basis functions

An explicit Galerkin method is formulated by using rational basis functions. The characteristics of the rational difference scheme are investigated with regard to consistency, stability and numerical convergence of the method. Numerical results are also presented.