Multi-Adaptive Galerkin Methods for ODEs I

We present multi-adaptive versions of the standard continuous and discontinuous Galerkin methods for ODEs. Taking adaptivity one step further, we allow for individual time-steps, order and quadrature, so that in particular each individual component has its own time-step sequence. This paper contains a description of the methods, an analysis of their basic properties, and a posteriori error analysis. In the accompanying paper [A. Logg, SIAM J. Sci. Comput., 27 (2003), pp. 741-758], we present adaptive algorithms for time-stepping and global error control based on the results of the current paper.