Found Comput Math (2014) 14:863–912 DOI 10.1007/s10208-013-9183-7 Convergence Analysis of Spatially Adaptive Rothe Methods Petru A. Cioica · Stephan Dahlke · Nicolas Döhring · Ulrich Friedrich · Stefan Kinzel · Felix Lindner · Thorsten Raasch · Klaus Ritter · René L. Schilling Received: 18 August 2012 / Revised: 18 October 2013 / Accepted: 27 November 2013 / Published online: 19 February 2014 © SFoCM 2014 Abstract This paper is concerned with the convergence analysis of the horizontal method of lines for evolution equations of the parabolic type. Following a semidis- cretization 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 Communicated by Andrew Stuart. P. A. Cioica · S. Dahlke · U. Friedrich · S. Kinzel (B) FB Mathematik und Informatik, AG Numerik/Optimierung, Philipps-Universität Marburg, Hans-Meerwein-Strasse, 35032 Marburg, Germany e-mail: kinzel@mathematik.uni-marburg.de P. A. Cioica e-mail: cioica@mathematik.uni-marburg.de S. Dahlke e-mail: dahlke@mathematik.uni-marburg.de U. Friedrich e-mail: friedrich@mathematik.uni-marburg.de N. Döhring · F. Lindner · K. Ritter Fachbereich Mathematik, TU Kaiserslautern, Postfach 3049, 67653 Kaiserslautern, Germany e-mail: doehring@mathematik.uni-kl.de F. Lindner e-mail: lindner@mathematik.uni-kl.de K. Ritter e-mail: ritter@mathematik.uni-kl.de T. Raasch Institut für Mathematik, AG Numerische Mathematik, Johannes Gutenberg-Universität Mainz, Staudingerweg 9, 55099 Mainz, Germany e-mail: raasch@uni-mainz.de 123