A Family of $H(div)$ Finite Element Approximations on Polygonal Meshes

Abstract

In this paper, we present a family of H(div)-compatible finite element spaces on strictly convex n-gons, whose construction makes use of generalized barycentric coordinates. In particular, for integers 0 ≤ k ≤ 2, we define finite element spaces with edge degrees of freedom that include polynomial vector fields of order k and whose vector fields have piecewise kth-order polynomial normal traces along the element boundary. These spaces suffer from the shortcoming that the image of the divergence operator includes nonpolynomial functions and, as such, their direct use in a mixed setting along with polynomial scalar fields may lead to unstable discretizations exhibiting degraded or no convergence. We present a general remedy for restoration of optimal convergence that involves "polynomial corrections" of vector fields and their divergence at the element level. These corrections are consistent with one another and require computation of suitable polynomial projection maps. In addition to the theoretical discussion, the performance of the discretization schemes based on the proposed spaces and the accompanying correction maps is numerically evaluated through their application to H(div) eigenvalue problems, mixed and least-squares approximations of linear and nonlinear porous media flow problems. Introduction.Motivated by the challenges of generating meshes for complex and evolving domains, the development of discretization schemes for solving partial differential equations on arbitrary polygonal and polyhedral meshes has become an active area of research in the past decade. The extension of the finite element method in this direction has been primarily focused on investigation and application of H 1 -conforming spaces involving vertex-based elements [37,36,35,22,39]. This relatively narrow scope can be attributed to the availability and ongoing development of the so-called generalized barycentric coordinates which serve as the underlying basis functions of these finite element spaces. The aim of the present work is to make use of barycentric coordinates to construct H(div)-compatible finite element spaces on convex polygonal partitions. These schemes can be employed for solving variational problems involving the H(div) space including porous media flow or mixed formulations of elasticity and Stokes flow. Since H(curl) in two dimensions is isomorphic to H(div) through a rotation, the present work can also be used to obtain discretizations for problems set in H(curl) such as Maxwells equations.A handful of edge-based finite element formulations are available in the literature for polygons. Kuznetsov and Repin [28] proposed a macroelement formulation using Raviart-Thomas spaces on a simplicial partition of the element. Euler, Schuhmann, and Weiland [19] and more recently Gillette, Rand, and Bajaj [23] have explored

Discontinuous Galerkin method with arbitrary polygonal finite elements

“…Afterwards a series of other papers have appeared concerning the FEM on polygonal or polyhedral meshes, e.g. [33,31,34,32,24,40,29,21,20,5,36,35]. There are two other methods that deals with polygonal/polyhedral meshes i.e.…”