Nearly H¹-optimal finite element methods

We examine the problem of finding the H¹ projection onto a finite element space of an unknown field satisfying a specified boundary value problem. Solving the projection problem typically requires knowing the exact solution. We circumvent this issue and obtain a Petrov-Galerkin formulation which achieves H¹ optimality. Requiring weighting functions to be defined locally on the element level permits only approximate H¹ optimality in multi-dimensional configurations. We investigate the relation between our formulation and other stabilized FEM formulations. We show, in particular, that our formulation leads to a derivation of the SUPG method. In special cases, the present formulation reduces to that of residual-free bubbles. Finally, we present guidelines for obtaining the Petrov weight functions, and include a numerical example for the Helmholtz equation.