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

Abstract

In the two parts of the present note we discuss several questions concerning the implementation of overdetermined least-squares collocation methods for higher index differential-algebraic equations (DAEs). Since higher index DAEs lead to ill-posed problems in natural settings, the discrete counterparts are expected to be very sensitive, which attaches particular importance to their implementation. In the present Part 1, we provide a robust selection of basis functions and collocation points to design the discrete problem. We substantiate a procedure for its numerical solution later in Part 2. Additionally, in Part 1, a number of new error estimates are proven that support some of the design decisions.

On the sensitivity of implementations of a least-squares collocation method for linear higher-index differential-algebraic equations

“…Even our first attempts showed surprisingly accurate results when applying the method to some linear examples [13]. More recently, we investigated the algorithmic ingredients of the method in more detail [10,11]. Not surprisingly, the basis representation and the choice of the integration nodes showed an important influence on the accuracy of the method.…”

“…In particular, qualitative and quantitative estimations for the condition numbers and norms of the representation map are proven for bases whose usefulness in the present applications has been established earlier [10,11].…”

“…In this section, we repeat the problem setting from [10] for the readers convenience. Consider a linear boundary-value problem for a DAE with properly involved derivative, A(t)(Dx) ′ (t) + B(t)x(t) = q(t), t ∈ [a, b],…”

“…Therefore, careful approximations of the integral in Φ are required. We take over the options provided in [10], in which M ≥ N +1 so-called collocation points…”

“…As detailed in [10], we have different options for the positive definite symmetric matrix L ∈ R M ×M , namely…”