Recovered finite element methods on polygonal and polyhedral meshes

¹ School of Mathematics, Cardiff University, Cardiff CF24 4AG, UK
² Department of Mathematics, University of Leicester, University Road, Leicester LE1 7RH, UK
³ School of Applied Mathematical and Physical Sciences, National Technical University of Athens, Zografou 15780, Greece
⁴ Department of Mathematical Sciences, University of Bath Claverton, Down Bath BA2 7AY, UK

^* Corresponding author: tmp38@bath.ac.uk, t.pryer@reading.ac.uk

Received: 12 July 2018
Accepted: 27 June 2019

Abstract

Recovered Finite Element Methods (R-FEM) have been recently introduced in Georgoulis and Pryer [Comput. Methods Appl. Mech. Eng. 332 (2018) 303–324]. for meshes consisting of simplicial and/or box-type elements. Here, utilising the flexibility of the R-FEM framework, we extend their definition to polygonal and polyhedral meshes in two and three spatial dimensions, respectively. An attractive feature of this framework is its ability to produce arbitrary order polynomial conforming discretizations, yet involving only as many degrees of freedom as discontinuous Galerkin methods over general polygonal/polyhedral meshes with potentially many faces per element. A priori error bounds are shown for general linear, possibly degenerate, second order advection-diffusion-reaction boundary value problems. A series of numerical experiments highlight the good practical performance of the proposed numerical framework.