Hermite analogs of the lowest order Raviart-Thomas mixed method for convection-diffusion equations
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 different senses, in the light of numerical experiments reported in Section 6. In this section, some numerical results, obtained using the methods described in Sections 3 and 5 for two test problems, are given. In Section 7, some comments on the whole work are made.
scientific article; zbMATH DE number 6973236
scientific article