Overdetermined least-squares collocation for higher-index differential-algebraic equations

Roswitha März (Institut für Mathematik, Humboldt-Universität zu Berlin)

R 3.28 Mon Z2 17:30-17:55

This is again a joint effort with Michael Hanke (KTH Stockholm) and ties in with the results we both presented at NUMDIFF-15.

We are looking for an approximate solution xπXπ of the initial- or boundary-value problem f((Dx)(t),x(t),t)=0,t[a,b],g(x(a),x(b))=0. The DAE in it can be of arbitrarily high index. The ansatz-space Xπ consists of componentwise and piecewise polynomial functions xπ on the grid π:a=t0<t1<<tn=b, with continuously connected part Dxπ. We use polynomials of degree N>1 for the component Dxπ but for the nondifferentiated part degree N1. Introducing MN+1 so-called collocation nodes 0τ1<<τM1 and in turn tji=tj1+τihj, we form the overdetermined collocation system f((Dxπ)(tji),xπ(tji),tji)=0,i=1,M,j=1,n,g(xπ(a),xπ(b))=0, which is then solved into a special least-squares sense for xπ. The procedure is inherently simple, the numerical tests are surprisingly good, but the underlying theory is quite demanding. Considering the fact that we are dealing here with an essentially ill-posed problem, it is important to implement it very carefully. Many questions are still open. We describe achievements, difficulties and surprises.

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