Adjoint Recovery of Superconvergent Linear Functionals from Galerkin Approximations. The One-dimensional Case

In this paper, we extend the adjoint error correction of Pierce and Giles (SIAM Rev. 42, 247–264 (2000)) for obtaining superconvergent approximations of functionals to Galerkin methods. We illustrate the technique in the framework of discontinuous Galerkin methods for ordinary differential and convection–diffusion equations in one space dimension. It is well known that approximations to linear functionals obtained by discontinuous Galerkin methods with polynomials of degree k can be proven to converge with order 2k + 1 and 2k for ordinary differential and convection–diffusion equations, respectively. In contrast, the order of convergence of the adjoint error correction method can be proven to be 4k + 1 and 4k, respectively. Since both approaches have a computational complexity of the same order, the adjoint error correction method is clearly a competitive alternative. Numerical results which confirm the theoretical predictions are presented.