Finite Difference Methods For Laplace

Finite Difference Methods for Laplace and Poisson Equation: The boundary value problems governed by the given partial differential equations along with suitable boundary conditions.

(a) Laplace’s equation: u_xx u_yy = ∇²u = 0, with u(x, y) prescribed on the boundary, that is, u(x, y) = f(x, y) on the boundary.

(b) Poisson’s equation: u_xx u_yy = ∇²u = G(x, y), with u(x, y) prescribed on the boundary, that is, u(x, y) = g(x, y) on the boundary.

In both the problems, the boundary conditions are called Dirichlet boundary conditions and the boundary value problem is called a Dirichlet boundary value problem.

 

Finite difference method We have a two dimensional domain (x, y) ∈ R. We superimpose on this domain R, a rectangular network or mesh of lines with step lengths h and k respectively parallel to the x- and y-axis. The mesh of lines is called a grid. The points of intersection of the mesh lines are called nodes or grid points or mesh points. The grid points are given by (x_i, y_j), (see Figs. 1.1 a, b), where the mesh lines are defined by

If h = k, then we have a uniform mesh. Denote the numerical solution at (x_i, y_j) by u_{i, j .}

                                                                                    

                                            Figure 1.1 (a) : Nodes in a rectangle.                          Figure :1.1 (b) Nodes in a square.

 

At the nodes, the partial derivatives in the differential equation are replaced by suitable difference approximations. That is, the partial differential equation is approximated by a difference equation at each nodal point. This procedure is called discretization of the partial differential equation. We use the following central difference approximations.

Solution of Laplace’s equation :

We apply the Laplace’s equation at the nodal point (i, j). Inserting the above approximations in the Laplace’s equation, we obtain

...................1.1

or,

.........................1.2

If h = k, that is, p = 1 (called uniform mesh spacing), we obtain the difference approximation as

................1.3

This approximation is called the standard five point formula. We can write this formula as

....................1.4

We observe that ui, j is obtained as the mean of the values at the four neighbouring points in the x and y directions.