This book has developed from lectures that the author gave for
mathematics students at the Ruhr-Universitat Bochum and the
Christian-Albrechts-Uni- versitat Kiel. This edition is the result of
the translation and correction of the German edition entitled Theone und
Numenk elliptischer Differential- gleichungen. The present work is
restricted to the theory of partial differential equa- tions of elliptic
type, which otherwise tends to be given a treatment which is either too
superficial or too extensive. The following sketch shows what the
problems are for elliptic differential equations. A: Theory of B:
Discretisation: c: Numerical analysis elliptic Difference Methods,
convergence, equations finite elements, etc. stability Elliptic Discrete
boundary value equations f-------- ----- problems E: Theory of D:
Equation solution: iteration Direct or with methods iteration methods
The theory of elliptic differential equations (A) is concerned with
ques- tions of existence, uniqueness, and properties of solutions. The
first problem of VI Foreword numerical treatment is the description of
the discretisation procedures (B), which give finite-dimensional
equations for approximations to the solu- tions. The subsequent second
part of the numerical treatment is numerical analysis (0) of the
procedure in question. In particular it is necessary to find out if, and
how fast, the approximation converges to the exact solution.