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

Abstract

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.