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Potential Analysis

, Volume 44, Issue 3, pp 473–495 | Cite as

On the Convergence Analysis of the Inexact Linearly Implicit Euler Scheme for a Class of Stochastic Partial Differential Equations

  • P. A. Cioica
  • S. Dahlke
  • N. Döhring
  • U. Friedrich
  • S. Kinzel
  • F. Lindner
  • T. Raasch
  • K. Ritter
  • R. L. Schilling
Article

Abstract

This paper is concerned with the adaptive numerical treatment of stochastic partial differential equations. Our method of choice is Rothe’s method. We use the implicit Euler scheme for the time discretization. Consequently, in each step, an elliptic equation with random right-hand side has to be solved. In practice, this cannot be performed exactly, so that efficient numerical methods are needed. Well-established adaptive wavelet or finite-element schemes, which are guaranteed to converge with optimal order, suggest themselves. We investigate how the errors corresponding to the adaptive spatial discretization propagate in time, and we show how in each time step the tolerances have to be chosen such that the resulting perturbed discretization scheme realizes the same order of convergence as the one with exact evaluations of the elliptic subproblems.

Keywords

Stochastic evolution equation Stochastic partial differential equation Euler scheme Rothe’s method Adaptive numerical algorithm Convergence analysis 

Mathematics Subject Classification (2010)

60H15 60H35 65M22 

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Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  • P. A. Cioica
    • 1
  • S. Dahlke
    • 1
  • N. Döhring
    • 2
  • U. Friedrich
    • 1
  • S. Kinzel
    • 1
  • F. Lindner
    • 2
  • T. Raasch
    • 3
  • K. Ritter
    • 2
  • R. L. Schilling
    • 4
  1. 1.FB Mathematik und Informatik, AG Numerik/OptimierungPhilipps-Universität MarburgMarburgGermany
  2. 2.Department of Mathematics, Computational Stochastics GroupTU KaiserslauternKaiserslauternGermany
  3. 3.Institut für Mathematik, AG Numerische MathematikJohannes Gutenberg-Universität MainzMainzGermany
  4. 4.FR Mathematik, Institut für Mathematische StochastikTU DresdenDresdenGermany

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