Recovered Finite Element Methods
For computational scientists using finite element methods, R-FEM offers a flexible framework that can produce stable conforming approximations and unify existing methods, though the novelty is incremental as it builds on known recovery techniques.
This paper introduces a new family of Galerkin finite element methods (R-FEM) that uses recovery operators over discontinuous approximation spaces, achieving optimal error bounds and unifying classical FEM and DG methods. Numerical experiments confirm good approximation properties.
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.