General Boundary-Value Problems

Abstract

Section 5.1 introduces the general elliptic linear differential equation of second order together with the Dirichlet boundary values. An important statement is the maximum-minimum principle in §5.1.2. In §5.1.3 sufficient conditions for the uniqueness of the solution and the continuous dependence on the data are proved. The discretisation of the general differential equation in a square is described in §5.1.4. Section 5.2 treats alternative boundary conditions replacing the Dirichlet data. Examples are the Neumann condition, the conormal derivative and the Robin boundary condition. Their discretisation (cf. §5.2.2) for general domains is rather laborious. Section 5.3 discusses differential equations of higher order. In particular, the biharmonic equation of fourth order is described in §5.3.1 followed by equations of order 2m in §5.3.2. The discretisation of the biharmonic equation is in §5.3.3.