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 Review, 42 (2000), pp. 247-264] 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 2 k + 1 and 2 k 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 4 k +1 and 4 k, 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.

• Type: Journal
• Year: 2007
• Where: JSCIC
• Authors: Bernardo Cockburn, Ryuhei Ichikawa
• Keywords: Adjoint Error Correction, Discontinuous Galerkin Methods, Galerkin Methods, JSCIC 2007