Summary
As is known [4]. theC o Galerkin solution of a two-point boundary problem using piecewise polynomial functions, hasO(h 2k) convergence at the knots, wherek is the degree of the finite element space. Also, it can be proved [5] that at specific interior points, the Gauss-Legendre points the gradient hasO(h k+1) convergence, instead ofO(h k). In this note, it is proved that on any segment there arek−1 interior points where the Galerkin solution is ofO(h k+2), one order better than the global order of convergence. These points are the Lobatto points.
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Bakker, M. A note onC o Galerkin methods for two-point boundary problems. Numer. Math. 38, 447–453 (1982). https://doi.org/10.1007/BF01396444
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DOI: https://doi.org/10.1007/BF01396444