Numerical Examples: Partial Differential Equations

Abstract

In this chapter, we will first show the numerical solution of the Dirichlet problem for a second order equation with nonconstant coefficients by the Ritz method. Then we give the solution of several modifications of this problem (the case of constant coefficients, the Neumann problem, the mixed problem). In the second part of this chapter, we shall solve — by the Ritz method, as before — a nonhomogeneous biharmonic equation with Dirichlet boundary conditions on a doubly connected domain. We will draw attention to the physical and technical interpretations of the presented problems and mention their solution by the Galerkin method. At the end of this chapter, we shall show, on a simple example, how to find an error estimate for the approximate solution.