The importance of adjoint consistency in the approximation of linear functionals using the discontinuous Galerkin finite element method

Abstract

We describe how a discontinuous Galerkin finite element method with interior penalty can be used to compute the solution to an elliptic partial differential equation and a linear functional of this solution can be evaluated. We show that, in order to have an adjoint consistent method and thus obtain optimal rates of convergence of the functional, a symmetric interior penalty Galerkin method must be used and, when the functional depends on the derivative of the solution of the PDE, an equivalent formulation of the functional must be used