Optimal Hessian recovery using a biorthogonal system with an application to adaptive refinement

Abstract

We present a method for recovering the Hessian from a linear finite element approach to achieve a higher rate of convergence. This method uses an \(L^{2}\)-based projection as well as boundary modification to achieve and improve the convergence rate. The projection uses a biorthogonal system to make the computation more numerically efficient. We present numerical examples to illustrate the efficiency and optimality of our approach on different meshes. The performance of our approach on adaptively refined meshes is briefly explored.