No 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.

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

Summary. B-convergence properties of defect correction methods based on the implicit Euler and midpoint schemes are discussed. The property of Bconvergence means that there exist global error bounds for nonlinear stiff problems independent of their stiffness. It turns out that the orders of Bconvergence of these methods coincide with the conventional orders of convergence of these methods derived under the assumption that h L is small (where L is a Lipschitz constant of the right-hand side). In Part I these assertions are reduced to the validity of the so-called Hypothesis A which is discussed in greater detail in Part II. Numerical experiments confirming the theoretical analysis are also given in Part II.

Summary. B-convergence properties of defect correction methods based on the implicit Euler and midpoint schemes are discussed. The property of Bconvergence means that there exist global error bounds for nonlinear stiff problems independent of their stiffness. It turns out that the orders of Bconvergence of these methods coincide with the conventional orders of convergence of these methods derived under the assumption that hL is small (where L is a Lipschitz constant of the right-hand side). In Part I these assertions are reduced to the validity of the so-called Hypothesis A which is discussed in greater detail in Part II. Numerical experiments confirming the theoretical analysis are also given in Part II.