Mixed approximation of elliptic problems on hybrid meshes
Ariel Lombardi, Universidad Nacional de Rosario and CONICET, Argentina
Sala conferenze IMATI-CNR, Pavia - Friday, October 6, 2017 h.16:30
Abstract. In this talk we consider the approximation in mixed form of elliptic problems in polyhedra. It is known that when the polyhedral domain is concave along edges or vertices, singularities may appear in the solution which degrade the numerical approximations. For the finite element method, strategies have been proposed in order to recover the optimal order of convergence, one of them being the use of meshes which are a priori adapted to the singularities. These meshes contain, in general, arbitrarily narrow elements, and as a consequence, the FEM tools to prove convergence have to manage this kind of elements. When mixed finite elements are considered, it is common the use of the H(div) -conforming Raviart-Thomas [Raviart-Thomas] spaces on tetrahedral meshes to approximate the vectorial variable, and then, interpolation error estimates for the Raviart-Thomas interpolation operator are one of the main tools to analyse the approximation error. In particular, anisotropic interpolation error estimates valid uniformly on anisotropic tetrahedral elements becomes needed to obtain optimal convergence results.
But, anisotropic error estimates are not uniformly valid on certain kind of anisotropic tetrahedra known as “slivers” which are needed to fill three dimensional spaces (see [Acosta et al.]).
For the simplest elliptic problem for the Laplace operator on a polyhedron, with the aim to avoid the presence of slivers in anisotropic meshes, we propose a generalization of the standard mixed method mentioned before, which, in particular, allows for the use of hybrid meshes made up of triangularly right prisms, tetrahedra and pyramids. The meshes can contain arbitrarily anisotropic right prisms in order to deal with edge singularities, and isotropic tetrahedra to be able to consider general polyhedral domains. And (isotropic) pyramids are needed in order to glue right prisms and tetrahedra.
For such a kind of meshes we introduce and analyse a mixed Finite/Virtual Element Method [Brezzi et al.]. The local discrete spaces coincide with the lowest order Raviart-Thomas spaces (and its extensions [Nedelec]) on tetrahedral and triangularly right prismatic elements, and extend it to pyramidal elements. The discrete scheme is well posed and optimal error estimates are proved on meshes which allow for anisotropic elements. In particular, local interpolation error estimates for the virtual space are optimal and anisotropic on anisotropic right prisms, which
can be used to obtain optimal approximation error estimates when the solution has edge or vertex singularities when suitably adapted meshes are used.
References
G. Acosta, Th. Apel, R.G. Durán, A.L. Lombardi. Error estimates for Raviart-Thomas interpolation of
any order on anisotropic tetrahedra. Math. Comp. 80\#273 (2011) 141--163.
F. Brezzi, R.S. Falk, L.D. Marini. Basic principles of virtual element methods. Math. Model. Numer. Anal. 48 (2014), 1227--1240.
J.C. Nédéléc. Mixed finite elements in R^3. Numer. Math. 35 (1980) 315--341.
P.A. Raviart, J.-M. Thomas. A mixed finite element method for second order elliptic problems, in Mathematical Aspects of the Finite Element Method, I. Galligani, E. Magenes, eds. Lectures Notes in Math. 606. Springer--Verlag 1977.
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