Special Differential Equations

Abstract

If the boundary-value problems have special properties, one often uses special discretisations for them. We give two examples. In Section 10.1 the principal part has jumping coefficients. Starting from the variational formulation, one obtains a strong formulation for each subdomain in which the coefficients are smooth. In addition, one gets transition equations at the inner boundary Υ. Finite-element methods should use a triangulation which follows Υ. Finally, in §10.1.4, we discuss the case that coefficients of terms different from the principal part are discontinuous. Typically the differential operators in fluid dynamics are nonsymmetric because of a derivative of first order. If this convections term becomes dominant, we obtain a singularly perturbed problem which is discussed in Section 10.2. In this case other discretisation variants are appropriate. In the case of difference method there is a conflict between stability and consistency conditions. Usual finite-element discretisation have similar difficulties. A remedy is the streamline-diffusion method explained in §10.2.3.2.