Adjoint recovery of superconvergent linear functionals from Galerkin approximations.

Abstract

The thesis is concerned with superconvergent approximations of linear functionals. We extend the adjoint error correction technique 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 problems in one dimension and two dimensions. In one dimension our focus is on ordinary differential and convection-diffusion equations. 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 can be proven to be 4 k + 1 and 4 k, using our technique. Since both approaches have a computational complexity of the same order, the adjoint error correction method is clearly a competitive alternative. In two dimensions we deal with a simple second-order elliptic model problem. We show that approximate functionals converge with order 4 k with our method. Numerical results which confirm the theoretical predictions are presented.