Multiadaptive Galerkin Methods for ODEs III: A Priori Error Estimates

TL;DR

This paper establishes general order a priori error estimates for multiadaptive Galerkin methods (mcG(q) and mdG(q)) used for solving initial value problems in ordinary differential equations, enhancing understanding of their accuracy.

Contribution

It provides the first comprehensive a priori error estimates for multiadaptive Galerkin methods, using dual solutions and residual representations to analyze error bounds.

Findings

Error estimates depend on the dual solution stability.

Order of convergence is established for the methods.

The approach links residuals with interpolation errors.

Abstract

The multiadaptive continuous/discontinuous Galerkin methods mcG(q) and mdG(q) for the numerical solution of initial value problems for ordinary differential equations are based on piecewise polynomial approximation of degree q on partitions in time with time steps which may vary for different components of the computed solution. In this paper, we prove general order a priori error estimates for the mcG(q) and mdG(q) methods. To prove the error estimates, we represent the error in terms of a discrete dual solution and the residual of an interpolant of the exact solution. The estimates then follow from interpolation estimates, together with stability estimates for the discrete dual solution.