Adjoint recovery of superconvergent functionals from approximate solutions of partial differential equations

Abstract

Motivated by applications in computational fluid dynamics, we present a method for obtaining estimates of integral functionals, such as lift or drag, that have twice the order of accuracy of the computed flow solution on which they are based. This is achieved through error analysis which uses an adjoint p.d.e. to relate the local errors in approximating the flow solution to the corresponding global errors in the functional of interest. Numerical evaluation of the local residual error together with an approximation solution to the adjoint equations may thus be combined to produce a correction for the computed functional value that yields twice the order of accuracy.

Numerical results are presented for the Poisson equation in one and two dimensions, and the nonlinear quasi-one-dimensional Euler equations. The superconvergence in these cases is as predicted by the a priori error analysis presented in the appendix. The theory is equally applicable to nonlinear equations in complex domains in multiple dimensions, and the technique has great potential for application in a range of engineering disciplines in which a few integral quantities are a key output of numerical approximations.