Interpolation Error Estimates For Vectorial Finite Elements On General Polyhedral Anisotropic Meshes

JAWTUSCHENKO, ALEXIS B.; LOMBARDI, ARIEL LUIS

Concepción

Workshop; Fifth Chilean Workshop on Nu- merical Analysis of Partial Differential Equations (WONAPDE); 2016

Facultad de Ingeniería Matemática, Universidad de Concepción

Abstract. In this talk we consider conforming approximations of H(div) and H(curl) by Raviart-Thomas and Nedelec finite elements, respectively, on anisotropic meshes. We are mainly interested in graded meshes which were proposed to recover the optimal order of convergence when the solution has singularities along edges or on vertices of a polyhedral domain. We review some known results on anisotropic interpolation error estimates for those finite elements on tetrahedral meshes [1, 2]. In particular we note that we can not obtain anisotropic error estimates for Raviart-Thomas elements on a kind of tetrahedra known as slivers. In this case the error estimates are uniform with respect to the anisotropy of the elements, but they are not of anisotropic type. Unfortunately, the slivers seems to be unavoidable in tetrahedral graded meshes. In order to overcome this difficulty, one possibility is to use graded meshes which combine tetrahedra, hexahedra, prisms, and pyramids. So we show interpolation error estimates valid on anisotropic prisms and pyramids the for vectorial elements mentioned before. Finally, using those results, an optimal convergence result is proved for a mixed formulation of the Poisson problem on a polyhedron when edge singularities are present. [1] Gabriel Acosta, Thomas Apel, Ricardo G. Durán, Ariel L. Lombardi. Error estimates for Raviart-Thomas interpolation of any order on anisotropic tetrahedra. Math. Comp. 80 #273 (2011) 141?163. [2] Ariel L. Lombardi. Interpolation error estimates for edge elements on anisotropic meshes. IMA J. Numer. Anal. 31 #4 (2011) 1683-1712.