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Journal of Scientific Computing
Erschienen in: Journal of Scientific Computing 3/2013

01.09.2013

Superconvergence of Any Order Finite Volume Schemes for 1D General Elliptic Equations

verfasst von: Waixiang Cao, Zhimin Zhang, Qingsong Zou

Erschienen in: Journal of Scientific Computing | Ausgabe 3/2013

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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 cases, the convergence rate can reach \(h^{r+2}\) and even \(h^{2r}\), comparing with \(h^{r+1}\) rate of the counterpart finite element method. Here \(r\) is the polynomial degree of the trial space. All theoretical results are justified by numerical tests.
Vorheriger Artikel A High-Order Kernel Method for Diffusion and Reaction-Diffusion Equations on Surfaces
Nächster Artikel Finding Multiple Solutions to Elliptic PDE with Nonlinear Boundary Conditions

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Metadaten
Titel
Superconvergence of Any Order Finite Volume Schemes for 1D General Elliptic Equations
verfasst von
Waixiang Cao
Zhimin Zhang
Qingsong Zou
Publikationsdatum
01.09.2013
Verlag
Springer US
Erschienen in
Journal of Scientific Computing / Ausgabe 3/2013
Print ISSN: 0885-7474
Elektronische ISSN: 1573-7691
DOI
https://doi.org/10.1007/s10915-013-9691-2

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