The Raviart–Thomas mixed finite-element method of the lowest order (Raviart and Thomas in mixed finite-element methods for second-order elliptic problems, Lecture Notes in mathematics, Springer, New York, 1977) commonly known as the \(RT_0\) method, is a well-established and popular numerical tool to solve diffusion-like problems providing flux continuity across inter-element boundaries. Douglas and Roberts extended the method to the case of more general second-order boundary-value problems including the convection–diffusion equations (cf. this journal Douglas in Comput Appl Math 1:91–103; 1982). The main drawback of these methods, however, is the poor representation of the primal variable by piecewise constant functions. The Hermite analog of the \(RT_0\) method for treating pure diffusion phenomena proposed in Ruas (J Comput Appl Math, 246:234–242; 2013) proved to be a valid alternative to attain higher order approximation of the primal variable while keeping intact the matrix structure and the quality of the discrete flux variable of the original \(RT_0\) method. Non-trivial extensions of this method are studied here that can be viewed as Hermite analogs of the two Douglas and Roberts’ versions of the \(RT_0\) method, to solve convection–diffusion equations. A detailed convergence study is carried out for one of the Hermite methods, and numerical results illustrate the performance of both of them, as compared to each other and to the corresponding mixed methods.
Convection–diffusion Douglas-Roberts Finite elements Hermite analog Lowest order Raviart–Thomas
Mathematics Subject Classification
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The first author is thankful for the financial support provided by CNPq through Grant 307996/2008-5 and the second author gratefully acknowledges the support of Statoil through the Akademia agreement.
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