Duality and finite elements

Post-processing in finite element analysis is essentially based on duality techniques: nodal values and point values in general or integrals of stresses are obtained by employing influence functions involving appropriate kernel functions Gih(y,x) which are the projections of the exact kernel functions Gi(y,x) onto the trial space Vh. Duality is therefore, as we want to show, a key concept of finite element methods. Questions of accuracy as well as questions of global and local equilibrium can be answered in this context. To this end we first formulate an Equivalence Theorem which establishes the connection between the finite element solution and the exact solution via Green's first identity. We next show that post-processing is essentially an application of Green's first or second identity. Finally, we formulate a new Projection Theorem which encapsulates the whole theory.