Convergence Analysis of Spatially Adaptive Rothe Methods

• Published in: Foundations of computational mathematics Vol. 14; no. 5; pp. 863 - 912
• Main Authors: Cioica, Petru A., Dahlke, Stephan, Döhring, Nicolas, Friedrich, Ulrich, Kinzel, Stefan, Lindner, Felix, Raasch, Thorsten, Ritter, Klaus, Schilling, René L.
• Format: Journal Article
• Language: English
• Published: Boston Springer US 01.10.2014; Springer; Springer Nature B.V
• Subjects: Analysis; Applications of Mathematics; Approximation; Asymptotic properties; Computer Science; Convergence; Convergence (Mathematics); Discretization; Economics; Linear and Multilinear Algebras; Math Applications in Computer Science; Mathematical analysis; Mathematical models; Mathematics; Mathematics and Statistics; Matrix Theory; Numerical Analysis; Original Paper; Preserves; Tolerances; United States; 65J08; 35K90; Adaptive wavelet methods; Besov spaces; 46E35; Horizontal method of lines; Parabolic evolution equations; 41A65; stage linearly implicit methods; 65M22; 65T60; Nonlinear approximation; 65M20
• ISSN: 1615-3375; 1615-3383
• DOI: 10.1007/s10208-013-9183-7

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.