Multi-Adaptive Galerkin Methods for ODEs I

TL;DR

This paper introduces multi-adaptive Galerkin methods for ODEs that allow individual components to have their own time-steps, orders, and quadratures, enhancing flexibility and efficiency in numerical solutions.

Contribution

It develops and analyzes multi-adaptive Galerkin methods for ODEs, enabling component-wise adaptivity in time-stepping, which was not previously available.

Findings

Methods have desirable stability and convergence properties.

Error analysis supports adaptive algorithm development.

Component-wise adaptivity improves computational efficiency.

Abstract

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.