SOME CURRENT QUESTIONS ON SOLVING LINEAR ELLIPTIC PROBLEMS

Common properties of elliptic problems and their solutions are compared with those of difference equations, with emphasis on the classical Dirichlet problem. Resistance network analogies are used to construct discrete harmonic functions; a new 9 POINT NET approximation is defined on 'elementary domains' whose solutions have O(h4) max-norm error. The convection-diffusion equation is studied with especial emphasis on new fast iterative methods for solving its difference approximations. A new analysis of the structure of the matrix representation of difference approximations reveals special features which can naturally be associated with diffusion and with convective processes. © 1990 Springer-Verlag.