Adjoint-Based, Superconvergent Galerkin Approximations of Linear Functionals

Abstract

We propose a new technique for computing highly accurate approximations to linear functionals in terms of Galerkin approximations. We illustrate the technique on a simple model problem, namely, that of the approximation of J(u), where $J(\cdot )$ is a very smooth functional and u is the solution of a Poisson problem; we assume that the solution u and the solution of the adjoint problem are both very smooth. It is known that, if $u_h$ is the approximation given by the continuous Galerkin method with piecewise polynomials of degree $k>0$, then, as a direct consequence of its property of Galerkin orthogonality, the functional $J(u_h)$ converges to J(u) with a rate of order $h^{2k}$. We show how to define approximations to J(u), with a computational effort about twice of that of computing $J(u_h)$, which converge with a rate of order $h^{4k}$. The new technique combines the adjoint-recovery method for providing precise approximate functionals by Pierce and Giles (SIAM Rev 42(2):247–264, 2000), which was devised specifically for numerical approximations without a Galerkin orthogonality property, and the accuracy-enhancing convolution technique of Bramble and Schatz (Math Comput 31(137):94–111, 1977), which was devised specifically for numerical methods satisfying a Galerkin orthogonality property, that is, for finite element methods like, for example, continuous Galerkin, mixed, discontinuous Galerkin and the so-called hybridizable discontinuous Galerkin methods. For the latter methods, we present numerical experiments, for $k=1,2,3$ in one-space dimension and for $k=1,2$ in two-space dimensions, which show that $J(u_h)$ converges to J(u) with order $h^{2k+1}$ and that the new approximations converges with order $h^{4k}$. The numerical experiments also indicate, for the p-version of the method, that the rate of exponential convergence of the new approximations is about twice that of $J(u_h)$.