Chapter 4 Elliptic Equations

In chapter 4, two finite difference schemes are built for Helmholtz equation with Dirich- let’s boundary value conditions. It is proved that the first scheme is O(h2 1 + h2 2) convergent in the norm of the Hilbert’s space H and in the maximum norm. The second scheme is O(h4 1 + h4 2 ) accurate and is also convergent in the norm of the Hilbert’s space H and in the maximum norm. Both schemes are solved by the Mathematica module. In the last section, Poisson’s equation with Dirichlet’s boundary conditions is solved by the method of lines. The chapter ends with a set of questions.