Recovered finite element methods

• Published in: Computer methods in applied mechanics and engineering Vol. 332; pp. 303 - 324
• Main Authors: Georgoulis, Emmanuil H., Pryer, Tristan
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier BV 15.04.2018
• Subjects: Approximation; Boundary value problems; Finite element analysis; Finite element method; Galerkin method; Mathematical analysis; Operators; Partial differential equations; Recovery
• ISSN: 0045-7825; 1879-2138
• DOI: 10.1016/j.cma.2017.12.026

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.