Convergence of the harmonic balance method for smooth Hilbert space valued differential-algebraic equations

TL;DR

This paper proves the convergence of the harmonic balance method for solving smooth Hilbert space valued DAEs, providing estimates based on Fourier series convergence and DAE structure, with applications in circuit modeling and structural dynamics.

Contribution

It establishes the first rigorous convergence analysis of harmonic balance for Hilbert space valued DAEs, including cases with known and unknown periods.

Findings

Convergence estimates depend on Fourier series convergence rate.

Both known and unknown period cases are analyzed.

Numerical experiments validate theoretical results.

Abstract

We analyze the convergence of the harmonic balance method for computing isolated periodic solutions of a large class of continuously differentiable Hilbert space valued differential-algebraic equations (DAEs). We establish asymptotic convergence estimates for (i) the approximate periodic solution in terms of the number of approximated harmonics and (ii) the inexact Newton method used to compute the approximate Fourier coefficients. The convergence estimates are deter-mined by the rate of convergence of the Fourier series of the exact solution and the structure of the DAE. Both the case that the period is known and unknown are analyzed, where in the latter case we require enforcing an appropriately defined phase condition. The theoretical results are illustrated with several numerical experiments from circuit modeling and structural dynamics.