In this paper, we present and study $ C^1 $ Petrov-Galerkin and Gauss collocation methods with arbitrary polynomial degree $ k $ ($ \ge 3 $) for one-dimen\-sional elliptic equations. We prove that, the solution and its derivative approximations converge with rate $ 2k-2 $ at all grid points; and the solution approximation is superconvergent at all interior roots of a special Jacobi polynomial of degree $ k+1 $ in each element, the first-order derivative approximation is superconvergent at all interior $ k-2 $ Lobatto points, and the second-order derivative approximation is superconvergent at $ k-1 $ Gauss points, with an order of $ k+2 $, $ k+1 $, and $ k $, 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 $ H^2 $, $ H^1 $, and $ L^2 $ norms. All theoretical findings are confirmed by numerical experiments.

中文翻译：

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一种 \ begin {document} $ C ^ 1 $ \ end {document} 一维一般椭圆问题和超收敛的Petrov-Galerkin方法和Gauss配置方法

在本文中，我们提出并研究一维椭圆形方程的任意多项式度为$ k $（$ \ ge 3 $）的$ C ^ 1 $ Petrov-Galerkin和高斯配置方法。我们证明，在所有网格点处，解及其导数近似均以速率k 2k-2 $收敛；并且在每个元素中阶k = 1的特殊Jacobi多项式的所有内部根的解逼近是超收敛的，在所有内部$ k-2 $ Lobatto点，一阶导数逼近是超收敛的，而第二阶导数逼近在$ k-1 $高斯点处是超收敛的，其阶数分别为$ k + 2 $，$ k + 1 $和$ k $。作为副产品，我们证明Petrov-Galerkin解和Gauss搭配解都朝着$ H ^ 2 $，$ H ^ 1 $和$ L ^ 2 $范数中精确解的特定Jacobi投影超收敛。所有的理论发现均通过数值实验得到证实。