有限体积法中2k猜想是否成立？
Is $2k$-Conjecture valid for finite volume methods?
AI总结：
本文研究了有限体积法的超收敛性质，证明了2k猜想，指出在网格顶点处，有限体积解在2k阶精度下逼近精确解，并通过数值实验验证了理论结果。
摘要：
本文研究了一类任意阶有限体积方法在矩形网格上的超收敛性质。主要结果是证明2k猜想：在底层矩形网格的每个顶点处，双k次有限体积解以O(h^{2k})的阶数逼近精确解。同时，还获得了有限体积离散误差在Lobatto和Gauss点处的超收敛性质。所有理论发现均通过数值实验得到验证。
英文摘要：
This paper is concerned with superconvergence properties of a class of finite volume methods of arbitrary order over rectangular meshes. Our main result is to prove {\it 2k-conjecture}: at each vertex of the underlying rectangular mesh, the bi-$k$ degree finite volume solution approximates the exact solution with an order $ O(h^{2k})$, where $h$ is the mesh size. As byproducts, superconvergence properties for finite volume discretization errors at Lobatto and Gauss points are also obtained. All theoretical findings are confirmed by numerical experiments.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
