Hermite analogs of the lowest order Raviart-Thomas mixed method for convection-diffusion equations
type: Item
instance of: scholarly article
title: Hermite analogs of the lowest order Raviart-Thomas mixed method for convection-diffusion equations
zbMATH Open document ID: 1412.65225
author: Vitoriano Ruas
author: F. A. Radu
published in: Computational and Applied Mathematics
publication date: 2018-11-05T00:00:00Z
review text: The Raviart-Thomas mixed finite element method, commonly known as the \(RT_0\) method, is a well-established numerical tool to solve diffusion-like problems. \textit{J. Douglas jun.} and \textit{J. E. Roberts} [Mat. Apl. Comput. 1, 91--103 (1982; Zbl 0482.65057)] extended the method to the case of more general second-order boundary value problems including the convection-diffusion equations. The Hermite analog of the \(RT_0\) method for studying pure diffusion phenomena is proposed by \textit{V. Ruas} [J. Comput. Appl. Math. 246, 234--242 (2013; Zbl 1262.76055)]. The reviewed work is devoted to the mathematical study of the Hermite finite element method proposed in \(\textit {V. Ruas}\) and \({P. Trales}\)' paper [``A Hermite finite element method for convection-diffusion equations'', In Simos T. et al.ed., AIP Proceedings of the 11th International Conference Numerical Analysis and Applied Mathematics, Rhodes, Greece (2013; \url{doi:10.1063/1.4825978 })], to solve the convection-diffusion equations. The paper is organized as follows. Section 1 is an introduction. In Section 2, some notations specify the model problem. The method proposed by Ruas and Trales [loc. cit.] is described in more detail in Section 3. In Section 4, these results which immediately lead to first-order error estimates are applied. Through duality techniques it is also proved that the method's convergence order in the \(L_2\)-norm is two, in contrast to the first-order ones that hold for the mixed extension of the \(RT_0\) method to convection-diffusion-reaction equations in non-divergence form and that is proposed by Douglas and Roberts [loc. cit.]. In Section 5, a variant of the method studied in Sections 3 and 4 is considered which can be viewed as the Hermite analog of the Douglas and Roberts mixed method [loc. cit.]. Both methods are compared, either to each other or to the corresponding Douglas and Roberts methods [loc. cit.], by checking their convergence properties and accuracy in differ…
Mathematics Subject Classification ID: 65N30
Mathematics Subject Classification ID: 65N12
Mathematics Subject Classification ID: 76R50
Mathematics Subject Classification ID: 76M10
zbMATH DE Number: 6973236
zbMATH Keywords: convection-diffusion
zbMATH Keywords: Douglas-Roberts
zbMATH Keywords: finite elements
zbMATH Keywords: Hermite analog
zbMATH Keywords: lowest order
zbMATH Keywords: Raviart-Thomas
reviewed by: Temur Jangveladze
MaRDI profile type: Publication
full work available at URL: s40314-017-0474-5
OpenAlex ID: W2732778001
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DOI: 10.1007/S40314-017-0474-5
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description: scientific article; zbMATH DE number 6973236