Superconvergence of Any Order Finite Volume Schemes for 1D General Elliptic Equations
Superconvergence of Any Order Finite Volume Schemes for 1D General Elliptic Equations
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DOI：
10.1007/s10915-013-9691-2
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发表时间：
2013-02
影响因子：
2.5
通讯作者：
Waixiang Cao;Zhimin Zhang;Q. Zou
中科院分区：
文献类型：
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作者：
Waixiang Cao;Zhimin Zhang;Q. Zou
给出并分析了一维椭圆型方程的任意阶有限体积格式。在该方案中，控制体是利用底层网格的子区间中的高斯点来构造的。给出了一个统一的证明，证明了有限体积格式在近似误差的能量和范数下具有最优的收敛速度。进一步证明了导数误差在所有高斯点处都是超收敛的，在某些特殊情况下，与同类有限元方法相比，收敛速度可以达到甚至。这是试验空间的多项式次。所有的理论结果都得到了数值试验的验证。
We present and analyze a finite volume scheme of arbitrary order for elliptic equations in the one-dimensional setting. In this scheme, the control volumes are constructed by using the Gauss points in subintervals of the underlying mesh. We provide a unified proof for the inf-sup condition, and show that our finite volume scheme has optimal convergence rate under the energy andnorms of the approximate error. Furthermore, we prove that the derivative error is superconvergent at all Gauss points and in some special cases, the convergence rate can reachand even, comparing withrate of the counterpart finite element method. Hereis the polynomial degree of the trial space. All theoretical results are justified by numerical tests.