A posteriori Error Estimation for the Spectral Deferred Correction Method

TL;DR

This paper develops an adjoint-based a posteriori error estimation technique for the spectral deferred correction method, enabling targeted error control and optimization of method parameters for solving ordinary differential equations.

Contribution

It introduces a novel error analysis framework using a nodally equivalent finite element method for spectral deferred correction, providing insights into error components and optimization strategies.

Findings

Derived an error formula split into error components.

Enabled optimal parameter adjustment for improved accuracy.

Validated the error estimation approach for spectral deferred correction.

Abstract

The spectral deferred correction method is a variant of the deferred correction method for solving ordinary differential equations. A benefit of this method is that is uses low order schemes iteratively to produce a high order approximation. In this paper we consider adjoint-based a posteriori analysis to estimate the error in a quantity of interest of the solution. This error formula is derived by first developing a nodally equivalent finite element method to the spectral deferred correction method. The error formula is then split into various terms, each of which characterizes a different component of the error. These components may be used to determine the optimal strategy for changing the method parameters to best improve the error.