Least-squares collocation for linear higher-index differential–algebraic equations

Abstract

We introduce a direct numerical treatment of nonlinear higher-index differential-algebraic equations by means of overdetermined polynomial least-squares collocation. The procedure is not much more computationally expensive than standard collocation methods for regular ordinary differential equations. The numerical experiments show impressive results. In contrast, the theoretical basic concept turns out to be considerably challenging. So far, quite recently convergence proofs for linear problems have been published. In the present paper we come up to a first convergence result for nonlinear problems. (Michael Hanke), maerz@math.hu-berlin.de (Roswitha März) 1 The number of unknowns equals the number of equations. 2 More precisely: Essentially ill-posed in Tichonov's sense, that is, the related operators feature nonclosed ranges.

Towards a reliable implementation of least-squares collocation for higher index differential-algebraic equations—Part 1: basics and ansatz function choices

“…The specific error estimation provided in [28] for the case of DAEs in Jordan chain form on equidistant grids may provide some further insight into the behavior of the instability threshold γ π . It is shown that…”

“…An overdetermined least-squares collocation method for the solution of boundary value problems for higher index differential-algebraic equations (DAEs) has been introduced in [28] and further investigated in [25][26][27]. A couple of sufficient convergence conditions have been established.…”

“…For the construction of a regularization method to treat an essentially ill-posed problem a Hilbert space setting of the problem is most convenient. For this reason, as in [26][27][28], we apply the spaces…”

“…where each λ S > 0 is the maximal eigenvalue of a certain symmetric, positive semidefinite matrix of size (S + 1) × (S + 1) [28,Lemma 3.3].…”

Collocation – A powerful tool for solving singular ODEs and DAEs

“…Since this approach is technically involved, it is worth to try to avoid it and provide a method which can be applied directly to the original DAE system of high index. At present, there are only some experimental results available, but they are quite encouraging and therefore, we shall briefly discuss them here [5]. To this aim, we restrict our attention to a models of index 3, (1).…”