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arXiv · 1207.0566

Any order superconvergence finite volume schemes for 1D general elliptic equations

Waixiang Cao·Zhimin Zhang·Qingsong Zou
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Abstract

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 and $L^2$ norms of the approximate error. Furthermore, we prove that the derivative error is superconvergent at all Gauss points and in some special case, the convergence rate can reach $h^{2r}$, where $r$ is the polynomial degree of the trial space. All theoretical results are justified by numerical tests.

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Waixiang Cao, Zhimin Zhang, Qingsong Zou. 2012-07-03. Any order superconvergence finite volume schemes for 1D general elliptic equations. https://arxiv.org/abs/1207.0566

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