Error estimates for Raviart-Thomas interpolation of any order on anisotropic tetrahedra

We prove optimal order error estimates for the Raviart-Thomas interpolation of arbitrary order under the maximum angle condition for triangles and under two generalizations of this condition, namely, the so-called three-dimensional maximum angle condition and the regular vertex property, for tetrahedra. Our techniques are different from those used in previous papers on the subject, and the results obtained are more general in several aspects. First, intermediate regularity is allowed; that is, for the Raviart-Thomas interpolation of degree k ≥ 0, we prove error estimates of order j + 1 when the vector field being approximated has components in WJ+1,p, for triangles or tetrahedra, where 0 ≤ j ≤ k and 1 ≤ p ≤ ∞. These results are new even in the two-dimensional case. Indeed, the estimate was known only in the case j = k. On the other hand, in the three-dimensional case, results under the maximum angle condition were known only for k = 0.Fil: Acosta, Gerardo Gabriel. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; ArgentinaFil: Apel, Thomas. Institut fur Mathematik und Bauinformatik, Universit at der Bundeswehr Munchen; ArmeniaFil: Duran, Ricardo Guillermo. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; ArgentinaFil: Lombardi, Ariel Luis. Universidad Nacional de General Sarmiento. Instituto de Ciencias; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentin

A nonconforming pressure-robust finite element method for the Stokes equations on anisotropic meshes

Most classical finite element schemes for the (Navier--)Stokes equations are neither pressure-robust, nor are they inf-sup stable on general anisotropic triangulations. A lack of pressure-robustness may lead to large velocity errors, whenever the Stokes momentum balance is dominated by a strong and complicated pressure gradient. It is a consequence of a method, which does not exactly satisfy the divergence constraint. However, inf-sup stable schemes can often be made pressure-robust just by a recent, modified discretization of the exterior forcing term, using H(div)-conforming velocity reconstruction operators. This approach has so far only been analyzed on shape-regular triangulations. The novelty of the present contribution is that the reconstruction approach for the Crouzeix--Raviart method, which has a stable Fortin operator on arbitrary meshes, is combined with results on the interpolation error on anisotropic elements for reconstruction operators of Raviart--Thomas and Brezzi--Douglas--Marini type, generalizing the method to a large class of anisotropic triangulations. Numerical examples confirm the theoretical results in a 2D and a 3D test case

A nonconforming pressure-robust finite element method for the Stokes equations on anisotropic meshes

Most classical finite element schemes for the (Navier-)Stokes equations are neither pressure-robust, nor are they inf-sup stable on general anisotropic triangulations. A lack of pressure-robustness may lead to large velocity errors, whenever the Stokes momentum balance is dominated by a strong and complicated pressure gradient. It is a consequence of a method, which does not exactly satisfy the divergence constraint. However, inf-sup stable schemes can often be made pressure-robust just by a recent, modified discretization of the exterior forcing term, using $\mathbf{H}(\operatorname{div})$-conforming velocity reconstruction operators. This approach has so far only been analyzed on shape-regular triangulations. The novelty of the present contribution is that the reconstruction approach for the Crouzeix-Raviart method, which has a stable Fortin operator on arbitrary meshes, is combined with results on the interpolation error on anisotropic elements for reconstruction operators of Raviart-Thomas and Brezzi-Douglas-Marini type, generalizing the method to a large class of anisotropic triangulations. Numerical examples confirm the theoretical results in a 2D and a 3D test case

The BEM with graded meshes for the electric field integral equation on polyhedral surfaces

We consider the variational formulation of the electric field integral equation on a Lipschitz polyhedral surface $\Gamma$. We study the Galerkin boundary element discretisations based on the lowest-order Raviart-Thomas surface elements on a sequence of anisotropic meshes algebraically graded towards the edges of $\Gamma$. We establish quasi-optimal convergence of Galerkin solutions under a mild restriction on the strength of grading. The key ingredient of our convergence analysis are new componentwise stability properties of the Raviart-Thomas interpolant on anisotropic elements

Anisotropic Raviart-Thomas interpolation error estimates using a new geometric parameter

We present precise Raviart-Thomas interpolation error estimates on anisotropic meshes. The novel aspect of our theory is the introduction of a new geometric parameter of simplices. It is possible to obtain new anisotropic Raviart-Thoma error estimates using the parameter. We also include corrections to an error in "General theory of interpolation error estimates on anisotropic meshes" (Japan Journal of Industrial and Applied Mathematics, 38 (2021) 163-191), in which Theorem 3 was incorrect.Comment: 26 pages, 2 figures. arXiv admin note: text overlap with arXiv:2106.0333