Symmetric marching technique and elliptic equations with variable coefficient involving mixed partial deriavatives

The symmetric marching technique developed in [1,2] has been extended to the elliptic equations with variable coefficients involving mixed partial derivatives. The restricted class of such equations of the form u_xx+2B(x, y)u_xy+c(x,y)u_yy=0 have been considered. Numerical results of model problems solved are presented.     

Projection methods for the numerical solution of non-self-adjoint elliptic partial differential equations

We compare the relative performances of two iterative schemes based on projection techniques for the solution of large sparse nonsymmetric systems of linear equations, encountered in the numerical solution of partial differential equations. The Block–Symmetric Successive Over-Relaxation (Block-SSOR) method and the Symmetric–Kaczmarz method are derived from the simplest of projection methods, that is, the Kaczmarz method. These methods are then accelerated using the conjugate gradient method, in order to improve their convergence. We study their behavior on various test problems and comment on the conditions under which one method would be better than the other. We show that while the conjugate-gradient-accelerated Block-SSOR method is more amenable to implementation on vector and parallel computers, the conjugate-gradient accelerated Symmetric–Kaczmarz method provides a viable alternative for use on a scalar machine.     

Analysis of Generalised Galerkin methods in the numerical solution of elliptic equations

The Petrov-Galerkin projection method is outlined for the solution of the linear elliptic equation Lu = f with homogeneous boundary conditions. By choosing appropriate finite dimensional trial and test spaces, the methods of weighted residuals, collocation, and H¹ Galerkin can be interpreted within the Petrov-Galerkin projection method framework. The important question of how best to choose the trial and test functions to suit a particular type of problem is then discussed. Objective criteria associated with the matrix which governs the Petrov-Galerkin numerical process are proposed.     

On numerical solution of stochastic partial differential equations of elliptic type

We study lattice approximations of stochastic PDEs of elliptic type, driven by a white noise on a bounded domain in ? ^d , for d = 1, 2, 3. We obtain estimates for the rate of convergence of the approximations.     

Symmetric marching technique for the Poisson equation. II. mixed boundary conditions

In paper I a symmetric marching technique for the discretized Poisson equation with Dirichlet boundary conditions was developed. In this paper, the symmetric marching technique is extended to cover mixed boundary value problems for Poisson equation. The results of some numerical experiments are also presented.     

Symmetric marching technique for the poisson equation. I. Dirichlet boundary conditions

The symmetric marching technique has been developed to solve the Poisson equation with Dirichlet boundary conditions. The method has been combined with a mesh refinement technique, which is used as an appropriate interpolation scheme to obtain a solution of the problem on finer grids. The effectiveness of the method has been demonstrated by solving some test examples.     

The modified alternating direction preconditioning method for the numerical solution of the elliptic self-adjoint second order and biharmonic equations

This paper introduces a generalised ADI method called the Modified Alternating Direction Preconditioning method (MADP method) for the numerical solution of the elliptic self-adjoint second order and biharmonic equations in a rectangle. The related theory of the MADP method is developed in detail and optimum values for the involved parameters are determined for both cases.     

The complexity of numerical methods for elliptic partial differential equations

We consider three Ritz-Galerkin procedures with Hermite bicubic, bicubic spline and linear triangular elements for approximating the solution of self-adjoint elliptic partial differential equations and a collocation with Hermite bicubics method for general linear elliptic equations defined on general two dimensional domains with mixed boundary conditions. We systematically evaluate these methods by applying them to a sample set of problems while measuring various performance criteria. The test data suggest that collocation is the most efficient method for general use.     

Positive solution for a class of nonlocal elliptic equations

In this paper, we devote ourselves to investigating the existence of positive solution for a class of nonlocal elliptic equations. Our approach is based on the fixed point index theory.     

Wavelets collocation methods for the numerical solution of elliptic BV problems

Based on collocation with Haar and Legendre wavelets, two efficient and new numerical methods are being proposed for the numerical solution of elliptic partial differential equations having oscillatory and non-oscillatory behavior. The present methods are developed in two stages. In the initial stage, they are developed for Haar wavelets. In order to obtain higher accuracy, Haar wavelets are replaced by Legendre wavelets at the second stage. A comparative analysis of the performance of Haar wavelets collocation method and Legendre wavelets collocation method is carried out. In addition to this, comparative studies of performance of Legendre wavelets collocation method and quadratic spline collocation method, and meshless methods and Sinc–Galerkin method are also done. The analysis indicates that there is a higher accuracy obtained by Legendre wavelets decomposition, which is in the form of a multi-resolution analysis of the function. The solution is first found on the coarse grid points, and then it is refined by obtaining higher accuracy with help of increasing the level of wavelets. The accurate implementation of the classical numerical methods on Neumann’s boundary conditions has been found to involve some difficulty. It has been shown here that the present methods can be easily implemented on Neumann’s boundary conditions and the results obtained are accurate; the present methods, thus, have a clear advantage over the classical numerical methods. A distinct feature of the proposed methods is their simple applicability for a variety of boundary conditions. Numerical order of convergence of the proposed methods is calculated. The results of numerical tests show better accuracy of the proposed method based on Legendre wavelets for a variety of benchmark problems.     

Duality estimates for the numerical solution of integral equations

Summary We formulate and prove Aubin-Nitsche-type duality estimates for the error of general projection methods. Examples of applications include collocation methods and augmented Galerkin methods for boundary integral equations on plane domains with corners and three-dimensional screen and crack problems. For some of these methods, we obtain higher order error estimates in negative norms in cases where previous formulations of the duality arguments were not applicable.     

一类高阶椭圆型方程的样条解法

从二元样条空间S2^1(△mn^(2))的理论出发，构造一类新的差分格式，并利用它得到了一类高阶椭圆型方程边值问题的样条解，并证明了这样的解的存在唯一性和收敛性问题．     

Averaging technique for the oscillation of second order damped elliptic equations

By using the averaging technique, we establish some oscillation theorems for the second order damped elliptic differential equation N↑∑↓i,j=1 Di[AIY（x）Djy]＋N↑∑↓i=1 bi（x）Diy＋c（x）f（y）=0 which extend and improve some known results in the literature.     

Block monotone iterative methods for numerical solutions of nonlinear elliptic equations

Summary. Two block monotone iterative schemes for a nonlinear algebraic system, which is a finite difference approximation of a nonlinear elliptic boundary-value problem, are presented and are shown to converge monotonically either from above or from below to a solution of the system. This monotone convergence result yields a computational algorithm for numerical solutions as well as an existence-comparison theorem of the system, including a sufficient condition for the uniqueness of the solution. An advantage of the block iterative schemes is that the Thomas algorithm can be used to compute numerical solutions of the sequence of iterations in the same fashion as for one-dimensional problems. The block iterative schemes are compared with the point monotone iterative schemes of Picard, Jacobi and Gauss-Seidel, and various theoretical comparison results among these monotone iterative schemes are given. These comparison results demonstrate that the sequence of iterations from the block iterative schemes converges faster than the corresponding sequence given by the point iterative schemes. Application of the iterative schemes is given to a logistic model problem in ecology and numerical ressults for a test problem with known analytical solution are given. Received August 1, 1993 / Revised version received November 7, 1994