A C^1 Petrov-Galerkin method and Gauss collocation method for 1D general elliptic problems and superconvergence

Abstract

<p style='text-indent:20px;'>In this paper, we present and study <inline-formula><tex-math id="M2">\begin{document}$ C^1 $\end{document}</tex-math></inline-formula> Petrov-Galerkin and Gauss collocation methods with arbitrary polynomial degree <inline-formula><tex-math id="M3">\begin{document}$ k $\end{document}</tex-math></inline-formula> (<inline-formula><tex-math id="M4">\begin{document}$ \ge 3 $\end{document}</tex-math></inline-formula>) for one-dimen\-sional elliptic equations. We prove that, the solution and its derivative approximations converge with rate <inline-formula><tex-math id="M5">\begin{document}$ 2k-2 $\end{document}</tex-math></inline-formula> at all grid points; and the solution approximation is superconvergent at all interior roots of a special Jacobi polynomial of degree <inline-formula><tex-math id="M6">\begin{document}$ k+1 $\end{document}</tex-math></inline-formula> in each element, the first-order derivative approximation is superconvergent at all interior <inline-formula><tex-math id="M7">\begin{document}$ k-2 $\end{document}</tex-math></inline-formula> Lobatto points, and the second-order derivative approximation is superconvergent at <inline-formula><tex-math id="M8">\begin{document}$ k-1 $\end{document}</tex-math></inline-formula> Gauss points, with an order of <inline-formula><tex-math id="M9">\begin{document}$ k+2 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M10">\begin{document}$ k+1 $\end{document}</tex-math></inline-formula>, and <inline-formula><tex-math id="M11">\begin{document}$ k $\end{document}</tex-math></inline-formula>, respectively. As a by-product, we prove that both the Petrov-Galerkin solution and the Gauss collocation solution are superconvergent towards a particular Jacobi projection of the exact solution in <inline-formula><tex-math id="M12">\begin{document}$ H^2 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M13">\begin{document}$ H^1 $\end{document}</tex-math></inline-formula>, and <inline-formula><tex-math id="M14">\begin{document}$ L^2 $\end{document}</tex-math></inline-formula> norms. All theoretical findings are confirmed by numerical experiments.