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Foundations of Computational Mathematics

, Volume 14, Issue 5, pp 863–912 | Cite as

Convergence Analysis of Spatially Adaptive Rothe Methods

  • Petru A. Cioica
  • Stephan Dahlke
  • Nicolas Döhring
  • Ulrich Friedrich
  • Stefan KinzelEmail author
  • Felix Lindner
  • Thorsten Raasch
  • Klaus Ritter
  • René L. Schilling
Original Paper

Abstract

This paper is concerned with the convergence analysis of the horizontal method of lines for evolution equations of the parabolic type. Following a semidiscretization in time by \(S\)-stage one-step methods, the resulting elliptic stage equations per time step are solved with adaptive space discretization schemes. We investigate how the tolerances in each time step must be tuned in order to preserve the asymptotic temporal convergence order of the time stepping also in the presence of spatial discretization errors. In particular, we discuss the case of linearly implicit time integrators and adaptive wavelet discretizations in space. Using concepts from regularity theory for partial differential equations and from nonlinear approximation theory, we determine an upper bound for the degrees of freedom for the overall scheme that are needed to adaptively approximate the solution up to a prescribed tolerance.

Keywords

Parabolic evolution equations Horizontal method of lines \(S\)-stage linearly implicit methods Adaptive wavelet methods Besov spaces Nonlinear approximation 

Mathematics Subject Classification

35K90 65J08 65M20 65M22 65T60 41A65 46E35 

Notes

Acknowledgments

This work was supported by the Deutsche Forschungsgemeinschaft (DFG, Grants DA 360/12-2, DA 360/13-2, RI 599/4-2, SCHI 419/5-2), a doctoral scholarship of the Philipps-Universität Marburg, and the LOEWE Center for Synthetic Microbiology (Synmikro), Marburg.

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Copyright information

© SFoCM 2014

Authors and Affiliations

  • Petru A. Cioica
    • 1
  • Stephan Dahlke
    • 1
  • Nicolas Döhring
    • 2
  • Ulrich Friedrich
    • 1
  • Stefan Kinzel
    • 1
    Email author
  • Felix Lindner
    • 2
  • Thorsten Raasch
    • 3
  • Klaus Ritter
    • 2
  • René L. Schilling
    • 4
  1. 1.FB Mathematik und Informatik, AG Numerik/OptimierungPhilipps-Universität MarburgMarburgGermany
  2. 2.Fachbereich MathematikTU KaiserslauternKaiserslauternGermany
  3. 3.Institut für Mathematik, AG Numerische MathematikJohannes Gutenberg-Universität MainzMainzGermany
  4. 4.Institut für Mathematische StochastikTU Dresden, FR MathematikDresdenGermany

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