Boundary-hybrid finite elements and a posteriori error estimation Original Research Article

Finite element methods that calculate the normal derivative of the solution along the mesh interfaces and recover the solution via local Neumann problems were introduced about two decades ago by I, Babus̆ka, J.T. Oden and J.K. Lee for the treatment of the homogeneous Laplace equation and called ‘boundary-hybrid method’. We revisit this approach for general symmetric and positive definite elliptic equations with homogeneous boundary conditions. The resulting approximation is nonconforming, and the corresponding error is orthogonal to all the conforming finite element subspaces. This crucial property shows immediately how to derive an a posteriori error estimator for conforming finite element approximations via Pythagorasʹ theorem. The investigation of this idea leads to a sound strategy for a posteriori error analysis which gives conservative, yet accurate, estimates, is cheap for good conforming approximations, and otherwise produces an enhanced solution at not significantly more than the cost normally expected for such a result.