7 - Numerical Solution of Elliptic Partial Differential Equations

Summary

Introduction

Elliptic partial differential equations are equations that have second derivatives in space but no time derivatives. The most important examples of elliptic PDEs are the Laplace equation, Poisson equation and Helmholtz equation. In two dimensions, the Laplace equation, Poisson equation, and Helmholtz equation are expressed as follows:

If the region of interest is a ≤ x ≤ b, and c ≤ y ≤ d, then assuming that one can discretize the region of interest into M subdivisions in x direction and N subdivisions in y direction, the grid size in the x and y directions are defined as Δx = (b-a)/M and Δy = (d - c)/N. Considering the Poisson equation and replacing the derivatives with central differences

The boundary conditions on all the four sides are supposed to be known as shown:

where g_a, g_b, g_c, and gd are known prescribed functions. The finite difference scheme is schematically shown in Figure 7.1 where the solution at the grid (I,j), ui, j is related to the solution at the four neighboring grid points (i+1.j), (i-1,j), (i,j+1), and (i,j-1).

Commonly occurring elliptic problems

Elliptic problems that manifest as elliptic PDEs, arise naturally in steady-state problems, i.e., in problems that are essentially concerned with states of equilibrium. That is, they arise in problems in which time t does not appear as an independent variable and, hence, where the initial values are not relevant. Hence, typical elliptic problems are essentially boundary value problems in which the boundary data are prescribed on given closed boundary curves (or surfaces/hypersurfaces for equations with more than two independent variables). The three most commonly occurring boundary conditions are (i) the Dirichlet problem, in which an elliptic PDE such as the Laplace equation, Equation (7.1), is solved with u being prescribed as a function of position along the whole boundary of the domain, (ii) the Neumann problem , in which an elliptic PDE such as the Laplace equation, Equation (7.1), is solved with being prescribed as a function of position along the whole boundary of the domain, where denotes the normal derivative along the boundary, and (iii) the Robin problem, in which an elliptic PDE such as the Laplace equation, Equation (7.1), is solved with prescribed along the whole boundary of the domain.