Joachim Nitsche

Error estimates for the numerical solution of elliptic differential equations

The numerical t rea tment of boundary-value problems for elliptic differential equations is often based on the difference method. The solution U produced by it, which is hoped to be an approximation to the exact solution u, is defined at the finitely many points of a grid and is obtained as the solution of an algebraic problem approximating the original boundary-value problem. The accuracy of this procedure will depend, of course, upon the fineness of the mesh. The estimation of the deviation U u in terms of this fineness is an important task. On the other hand, the quality of the error estimates obtainable finds its restrictions in the regularity properties of the solution or, what is the same, in the behavior of the given data. In what follows the bouffdary-value problem

Fehlerschranken bei Eigenwertproblemen normalisierbarer Matrizen

Die Vektoren mogen die Eigenvektoren einer normalisierbaren Matrix A approximieren und λi die Eigenwerte. Mit Hilfe der Abweichungen und des Mindestabstandes der λi werden Fehlerschranken fur die Genauigkeit von und λi hergeleitet, zu denen fur nach passender Normierung noch ein Faktor hinzutritt, der durch die Nichtorthogonalitat der Eigenvektoren bedingt ist. Let and λi be a set of approximate eigenvectors and eigenvalues of a normalizable matrix A. With the aid of and the minimum distance of the λi bounds for the error of and λi are derived, those for being supplemented, after suitable normalisation, by the factor resulting from the non-orthogonality of the eigenvectors.