AI 中文总结
该研究确定了一维样条有限元近似的自然超收敛点条件，推导了误差渐近展开式，并通过数值实验验证了理论结果。
AI 中文摘要
我们研究一维样条有限元近似的自然超收敛点与渐近展开。对于次数为k且光滑度满足0≤μ≤k-1的样条空间，我们证明：若在某点周围大小为Ch|ln h|的区域内网格对称，则当k-s为偶数时，误差的s阶导数会呈现阶数为O(h^{k+2-s})的增强收敛。该条件对于低导数阶数s=0,1的情况是最优的；本分析表明，相同的局部条件对所有容许的s均充分。此外，通过在勒让德多项式中展开误差，闭合定理结合伽辽金正交性与超收敛条件确定了主导阶勒让德系数（误差的渐近展开）。对于μ=k-1（B样条）和μ=k-2，伽辽金正交性条件消失，系数仅由超收敛条件确定。渐近展开可通过勒让德多项式上的简单原函数递推式表达，所得多项式的零点编码了所有导数阶数的完整超收敛点集合。对选定的(k,μ)组合进行的数值实验证实了理论预测。
英文摘要
We study the natural superconvergence points and asymptotic expansions of one-dimensional spline finite element approximations. For a spline space of degree $k$ and any smoothness $0\leμ\le k-1$, we prove that the $s$-th derivative of the error exhibits enhanced convergence of order $O(h^{k+2-s})$ at points where $k-s$ is even, provided the mesh is symmetric within a region of size $Ch|\ln h|$ around the point. This condition is known to be optimal for the cases of low derivative order $s=0,1$; the present analysis shows that the same local condition is sufficient for all admissible $s$. Moreover, by expanding the error in Legendre polynomials, a closure theorem determines the leading-order Legendre coefficients (the asymptotic expansion of the error) by combining the Galerkin orthogonality with the superconvergence conditions. For $μ=k-1$ (B-splines) and $μ=k-2$, the Galerkin orthogonality conditions vanish and the coefficients are determined solely by the superconvergence conditions. The asymptotic expansion can be expressed through a simple antiderivative recurrence on Legendre polynomials. The resulting polynomial's zeros encode the complete set of superconvergence points for all derivative orders. Numerical experiments for selected $(k,μ)$ pairs confirm the theoretical predictions.
Comments28 pages
