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<p>A numerical study of tetrahedral Raviart-Thomas mixed finite element methods is presented in the solution of model second order boundary value problems posed in a curved spatial domain. An emphasis is given to the case where normal fluxes are prescribed on a boundary portion. In this case the question on the best way to enforce known boundary degrees of freedom is raised. It seems intuitive that the normal component of the flux variable should preferably not take up corresponding prescribed values at nodes shifted to the boundary of the approximating polyhedron in the underlying normal direction. This is because an accuracy downgrade is to be expected, as shown in [<CitationRef CitationID="CR1">1</CitationRef>] and [<CitationRef CitationID="CR2">2</CitationRef>]. In the former work accuracy improvement is achieved by means of a standard Galerkin formulation with parametric elements. The latter one in turn advocates the use of straight-edged triangles combined with a Petrov-Galerkin formulation, in which the aforementioned shift applies only to the test-flux space, while the shape-flux space consists of fields whose fluxes satisfy the prescribed conditions on the true boundary. The first purpose of this article is to show that the method studied in [<CitationRef CitationID="CR2">2</CitationRef>] for two-dimensional problems can be extended quite naturally to the three-dimensional case. More particularly we illustrate this by carrying out numerical experimentation with such a version for the two lowest order methods of this family, as compared to the corresponding do-nothing strategy. In the case of the lowest order method this comparative study is enriched by assessing as well the performance of its Hermite analog introduced in [<CitationRef CitationID="CR8">8</CitationRef>].</p>

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Assessment of Variants of Raviart-Thomas Mixed Elements Handling Spatial Curved Domains with Straight-Edged Tetrahedra

  • 期刊论文

  • 发表日期：

    2026年3月18日

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  • Vitoriano Ruas

摘要

A numerical study of tetrahedral Raviart-Thomas mixed finite element methods is presented in the solution of model second order boundary value problems posed in a curved spatial domain. An emphasis is given to the case where normal fluxes are prescribed on a boundary portion. In this case the question on the best way to enforce known boundary degrees of freedom is raised. It seems intuitive that the normal component of the flux variable should preferably not take up corresponding prescribed values at nodes shifted to the boundary of the approximating polyhedron in the underlying normal direction. This is because an accuracy downgrade is to be expected, as shown in [1] and [2]. In the former work accuracy improvement is achieved by means of a standard Galerkin formulation with parametric elements. The latter one in turn advocates the use of straight-edged triangles combined with a Petrov-Galerkin formulation, in which the aforementioned shift applies only to the test-flux space, while the shape-flux space consists of fields whose fluxes satisfy the prescribed conditions on the true boundary. The first purpose of this article is to show that the method studied in [2] for two-dimensional problems can be extended quite naturally to the three-dimensional case. More particularly we illustrate this by carrying out numerical experimentation with such a version for the two lowest order methods of this family, as compared to the corresponding do-nothing strategy. In the case of the lowest order method this comparative study is enriched by assessing as well the performance of its Hermite analog introduced in [8].