# Recovered Finite Element Methods

## TL;DR

This paper introduces recovered finite element methods (R-FEM), a new family of Galerkin methods that combine features of classical and discontinuous Galerkin approaches, offering stability, optimal error bounds, and adaptability.

## Contribution

The paper presents a novel family of R-FEM that unifies and extends classical finite element and discontinuous Galerkin methods with proven stability and error estimates.

## Key findings

- R-FEM can produce stable conforming approximations.
- Optimal a priori error bounds are established.
- Numerical experiments demonstrate good approximation properties.

## Abstract

We introduce a family of Galerkin finite element methods which are constructed via recovery operators over element-wise discontinuous approximation spaces. This new family, termed collectively as recovered finite element methods (R-FEM) has a number of attractive features over both classical finite element and discontinuous Galerkin approaches, most important of which is its potential to produce stable conforming approximations in a variety of settings. Moreover, for special choices of recovery operators, R-FEM produces the same approximate solution as the classical conforming finite element method, while, trivially, one can recast (primal formulation) discontinuous Galerkin methods. A priori error bounds are shown for linear second order boundary value problems, verifying the optimality of the proposed method. Residual-type a posteriori bounds are also derived, highlighting the potential of R-FEM in the context of adaptive computations. Numerical experiments highlight the good approximation properties of the method in practice. A discussion on the potential use of R-FEM in various settings is also included.