Recovered finite element methods on polygonal and polyhedral meshes

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.

Recovered finite element methods

“…This way once can construct conforming approximations of elliptic problems on polytopic meshes where one has access to the whole of the approximate solution; this is in contrast to the recent virtual element framework [5] whereby only certain functionals of the approximate solution are available to the user. This is a significant topic in its own right and will be discussed in detail in [17].…”

“…Since the use of total degree bases on such meshes necessitates relaxation of conformity requirements for the approximation spaces, dG methods have been used in this context as the underlying discretization. It is evident that R-FEM can be naturally extended to accept such reduced approximation spaces, once a suitable choice of recovery is constructed; we refer to the forthcoming work [17] for the construction of R-FEM for general polygonal/polyhedral element shapes. Also, recalling that R-FEM is based on discontinuous approximation spaces also, R-FEM is expected to be able to produce stable conforming approximations in the context of convection-dominated problems.…”

A Posteriori Error Estimates for the Weak Galerkin Finite Element Methods on Polytopal Meshes

“…It can be achieved through adaptive mesh refinement that generates a mesh tailored in reducing computational errors at places of great need. Adaptive mesh refinement will be more local and effective for the finite element methods that allow general mesh [11,13]. In recent years, many numerical schemes have been developed and analyzed on general polytopal mesh such as HDG method, mimetic finite difference method, virtual element method and hybrid high-order method [5,6,10,22].…”

“…An important feature of the WG methods is allowing the use of general polytopal meshes [18,19,25,27]. The importance of such feature in adaptive finite element methods is well stated in [11,13].…”

On the numerical approximation of p‐biharmonic and ∞‐biharmonic functions

“…It has been well studied in the context of both C 1 finite elements and discontinuous Galerkin methods; for example, the papers study the use of h – k dG finite elements (where k here means the local polynomial degree as opposed to the usual convention which is p ) applied to the (2‐) Bilaplacian. Alternative methods do exist, including those of virtual element type and recovered element type . In addition to this, the classical work of proposed mixed methods for the linear problem whose analysis was based on the mesh‐dependent norms in Ref.…”