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(·) 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 uh 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(uh) converges to J(u) with a rate of order h2 k. We show how to define approximations to J(u), with a computational effort about twice of that of computing J(uh) , which converge with a rate of order h4 k. 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(uh) converges to J(u) with order h2 k + 1 and that the new approximations converges with order h4 k. 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(uh).
Bibliographical noteFunding Information:
Research supported by the U.S. National Science Foundation Grants DMS-1522657 and DMS-1522672.
- Adjoint-based error correction
- Approximation of linear functionals
- Galerkin methods