Discretization Error Estimates for Certain Splitting Procedures for Solving First Biharmonic Boundary Value Problems (original) (raw)
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The Dirichlet problem for biharmonic equation in a rectangular region is considered. The method of splitting is used and two classes of finite difference approximations are defined. Two semi-iterative procedures are considered for obtaining the solution of the resulting coupled system of algebraic equations. It is shown that the rate of convergence of the iterative procedures depends upon the choice of the difference approximation. Estimates for optimum iteration parameters are given and several comparisons are made. An attempt is made to unify the ideas on the splitting technique for solving the first biharmonic boundary value problem.
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The coefficients for a nine-point high-order accuracy discretization scheme for a biharmonic equation ∇ 4 u = 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 ∂ 2 u/∂n 2 or (2) u and ∂u/∂n (where ∂/∂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(h 6) on a square mesh (hx = hy = h) and of the fourth-order O(h 4 x , h 2 x h 2 y , h 4 y) 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.
Journal of Computational Physics, 2005
The Poisson equation subject to Dirichlet boundary conditions on an irregular domain can be treated by embedding the region in a rectangular domain and solving using finite differences over the rectangle. The crucial issue is the discretization of the boundaries of the irregular domain. In the past, both linear and quadratic boundary treatments have been used and error bounds have been derived in both cases, showing that the linear case gives uniform second-order accuracy, whereas the quadratic case gives third-order accuracy at the boundaries and second-order accuracy internally. Thus, it has been recommended that the linear boundary treatment be used, as it is simpler, gives rise to a symmetric matrix formulation and has uniform accuracy. The present work shows that this argument is inadequate, because the coefficients of the error terms also play an important role. We demonstrate this in the 1-D case by determining explicit expressions for the error for both the linear and quadratic boundary treatments. It is shown that for the linear case the coefficient of error is in general large enough to dominate the calculation and that therefore it is necessary to use a quadratic boundary treatment in order to obtain errors comparable with those obtained for a regular domain. We go on to show that the 1-D expressions for error can be used to approximate the boundary error for 2-D problems, and that for the linear treatment, the boundary error again dominates.
Finite Difference Method with Dirichlet Problems of 2D Laplace’s Equation in Elliptic Domain
Pakistan Journal of Engineering, Technology & Science, 2018
In this study finite difference method (FDM) is used with Dirichlet boundary conditions on rectangular domain to solve the 2D Laplace equation. The chosen body is elliptical, which is discretized into square grids. The finite difference method is applied for numerical differentiation of the observed example of rectangular domain with Dirichlet boundary conditions. The obtained numerical results arecompared with analytical solution. The obtained results show the efficiency of the FDM and settled with the obtained exact solution. The study objective is to check the accuracy of FDM for the numerical solutions of elliptical bodies of 2D Laplace equations. The study contributes to find the heat (temperature) distribution inside a regular rectangular elliptical discretized body.
Numerische Mathematik, 2001
The aim of this paper is to give a new method for the numerical approximation of the biharmonic problem. This method is based on the mixed method given by Ciarlet-Raviart and have the same numerical properties of the Glowinski-Pironneau method. The error estimate associated to these methods are of order O(h k−1) for k≥ 2. The algorithm proposed in this paper converges even for k≥ 1, without any regularity condition on ω or ψ. We have an error estimate of order O(h k) in case of regularity.
Numerical Analysis and Applications, 2018
The present paper uses a new two-level implicit difference formula for the numerical study of one-dimensional unsteady biharmonic equation with appropriate initial and boundary conditions. The proposed difference scheme is second-order accurate in time and third-order accurate in space on non-uniform grid and in case of uniform mesh, it is of order two in time and four in space. The approximate solutions are computed without using any transformation and linearization. The simplicity of the proposed scheme lies in its three-point spatial discretization that yields block tri-diagonal matrix structure without the use of any fictitious nodes for handling the boundary conditions. The proposed scheme is directly applicable to singular problems, which is the main utility of our work. The method is shown to be unconditionally stable for model linear problem for uniform mesh. The efficacy of the proposed approach has been tested on several physical problems, including the complex fourth-order nonlinear equations like Kuramoto-Sivashinsky equation and extended Fisher-Kolmogorov equation, where comparison is done with some earlier work. It is clear from numerical experiments that the obtained results are not only in good agreement with the exact solutions but also competent with the solutions derived in earlier research studies.