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
\(J(u_h)\).