International Journal of Mathematics and Mathematical Sciences
Volume 2 (1979), Issue 4, Pages 651-667
doi:10.1155/S016117127900048X
Uniform stability of linear multistep methods in Galerkin procedures for parabolic problems
Der Universitat Stuttgart, 7 Stuttgart 80, Pfaffenwaldring 57, Germany
Received 3 May 1978; Revised 3 April 1979
Copyright © 1979 Eckart Gekeler. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
Linear multistep methods are considered which have a stability region S and are D-stable on the whole boundary ∂S⊂S of S. Error estimates are derived which hold uniformly for the class of initial value problems Y′=AY+B(t), t>0, Y(0)=Y0 with normal matrix A satisfying the spectral condition Sp(ΔtA)⊂S, Δt time step, Sp(A) spectrum of A. Because of this property, the result can be applied to semidiscrete systems arising in the Galerkin approximation of parabolic problems. Using known results of the Ritz theory in elliptic boundary value problems error bounds for Galerkin multistep procedures are then obtained in this way.