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

Abstract

We show how the QDeC estimator, an efficient and asymptotically correct a-posteriori error estimator for collocation solutions to ODE systems, can be extended to differential-algebraic equations of index 1. Problem settingWe consider linear systems of DAEs of index 1,with appropriately smooth dataFor the purpose of this analysis, we assume that (1) is well-posed as an initial value problem, with smooth solution x * (t). We assume m > n, and kerConditions (2) imply that (AD)(t) ∈ R m×m is singular, with rank (AD)(t) ≡ n. 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 bywhere, andfor i = 0 . . . N −1, j = 1 . . . s. Note, in particular, that c s = 1 is essential for our analysis. We also assume that s is even, which will be necessary to guarantee the asymptotic correctness of our error estimator to be defined in Section 2. We also denote h := max i=0...N −1 h i . Defect-based error estimationIn the context of regular and singular ODEs, a method for computing an a-posteriori estimate of the global error e := p − x * was proposed in [2] and implemented in [1]. This method is based on the defect correction principle [6,7]. In particular, for a special realization of the defect, an efficient, asymptotically correct error estimator, the QDeC estimator, was designed and analyzed in [2,3] for collocation solutions on arbitrary grids. These ideas are now extended to the DAE context, which turns out not to be straightforward because of the coupling between differential and algebraic components. A naive application of the procedure proposed in [6] would be based on the pointwise defect

Error Estimation Techniques Based on Defect Computation and Global Reconstruction

“…The observed orders are m = 3 for ||ẽ|| ∞ and m +2 = 5 for ||δ || ∞ = ||ε −ẽ|| ∞ . For related results concerning implicit first order systems, in particular index 1 DAEs, see [5].…”