Defect-based a Posteriori Error Estimation for Index 1 DAEs with a Singularity of the First Kind

Abstract

A computationally efficient a posteriori error estimator is introduced and analyzed for collocation solutions to linear index-1 DAEs with properly stated leading term. The procedure is based on a modified defect correction principle, extending an established technique from the ODE context to the DAE case. We prove that the resulting error estimate is asymptotically correct, and illustrate the method by means of a numerical example. To keep the presentation reasonably self-contained, we also briefly review some arguments from the literature on DAEs concerning the decoupling of the problem and its discretization, which is essential for our analysis.

Numerical Treatment of Singular BVPs: The New MATLAB Code bvpsuite

“…As already mentioned, eigenvalue problems, see [3], [31], and differential algebraic equations, cf. [8], [25], are within the scope of our code, but it can also be applied in case of non-standard singularities. In [27], we investigated the following singular equation which originates from the theory of shallow membrane caps,…”

An efficient asymptotically correct error estimator for collocation solutions to singular index-1 DAEs

“…2.2. First results for case where the inherent ODE is regular has been discussed in [6]. The aim of the present paper is to extend these results to the case where the inherent ODE has a singularity of the first kind at t = 0.…”

“…Our focus is on the index-1 case with a singularity of the first kind. The regular index-1 case can be seen as a simple special case, which is also considered in [6]. with appropriately smooth data A(t) ∈ R m×n , D(t) ∈ R n×m , B(t) ∈ R m×m .…”

Defect‐based a‐posteriori error estimation for differential‐algebraic equations

“…The requirement that D(t) be constant is not a real restriction, as any such system with varying D(t) can be rewritten by introducing a new variable u(t) = D(t)x(t), resulting in a larger system of the type (1) forx(t) := (x(t), u(t)) T with a constant matrixD(t) ≡D, see [4]. We consider collocation solutions p(t) for (1), defined by…”

“…For the discrete systems (3) and (7) a similar decoupling argument is used. For details of the proof we refer to [4].…”