Convergence Analysis of Spatially Adaptive Rothe Methods
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- 作者：Petru A. Cioica (1)
Stephan Dahlke (1)
Nicolas D枚hring (2)
Ulrich Friedrich (1)
Stefan Kinzel (1)
Felix Lindner (2)
Thorsten Raasch (3)
Klaus Ritter (2)
Ren茅 L. Schilling (4) - 关键词：Parabolic evolution equations ; Horizontal method of lines ; $$S$$ S ; stage linearly implicit methods ; Adaptive wavelet methods ; Besov spaces ; Nonlinear approximation ; 35K90 ; 65J08 ; 65M20 ; 65M22 ; 65T60 ; 41A65 ; 46E35
- 刊名：Foundations of Computational Mathematics
- 出版年：2014
- 出版时间：October 2014
- 年：2014
- 卷：14
- 期：5
- 页码：863-912
- 全文大小：684 KB
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Stephan Dahlke (1)
Nicolas D枚hring (2)
Ulrich Friedrich (1)
Stefan Kinzel (1)
Felix Lindner (2)
Thorsten Raasch (3)
Klaus Ritter (2)
Ren茅 L. Schilling (4)
1. FB Mathematik und Informatik, AG Numerik/Optimierung, Philipps-Universit盲t Marburg, Hans-Meerwein-Strasse, 35032聽, Marburg, Germany
2. Fachbereich Mathematik, TU Kaiserslautern, Postfach 3049, 67653聽, Kaiserslautern, Germany
3. Institut f眉r Mathematik, AG Numerische Mathematik, Johannes Gutenberg-Universit盲t Mainz, Staudingerweg 9, 55099聽, Mainz, Germany
4. Institut f眉r Mathematische Stochastik, TU Dresden, FR Mathematik, 01062聽, Dresden, Germany - ISSN：1615-3383
文摘
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.