Abstract
New a posteriori (computable) upper bounds for theL 2-norms, both ofD(u−v) and ofu−v are proposed, whereu is the exact solution of the boundary value problem
andv any approximation of it (D is here the vector of partial derivatives with respect to the components ofx).
It is shown that the new error bounds are better than the classical one, which is proportional to ‖Av−f‖, in many cases. This happens, e. g., ifq has some zero point inG, as in the case of a Poisson equation.
Zusammenfassung
Neue a posteriori (d.h. berechenbare) obere Schranken fürD(u−v) undu−v werden vorgeschlagen, wobeiu die exakte Lösung des Randwertproblems
undv irgendeine Approximation dieser Lösung ist (D ist hier der Vektor der partiellen Ableitungen in Bezug auf die Komponenten vonx).
Es wird gezeigt, daß diese Fehlerschranken in vielen Fällen besser sind als die klassische Schranke, die proportional zu ‖Av−f‖ ist. Dies tritt etwa ein, wennq eine Nullstelle inG hat, z.B. im Fall der Poissonschen Gleichung.
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References
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Kioustelidis, J.B. L 2-Error bounds for approximate solutions of elliptic partial differential equations with Dirichlet-boundary conditions. Computing 43, 133–140 (1989). https://doi.org/10.1007/BF02241857
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DOI: https://doi.org/10.1007/BF02241857