338	RWTH Publication No: 47344        2012        IGPM338.pdf
TITLE	A Note on Adjoint Error Estimation for One-Dimensional Stationary Conservation Laws with Shocks
AUTHORS	Jochen Schütz, Sebastian Noelle, Christina Steiner, Georg May
ABSTRACT	We consider one-dimensional steady-state conservation laws with discontinuous solutions. Giles and Pierce [7] realized that a shock leads to a new term in the adjoint error representation for target functionals. This term disappears if and only if the adjoint solution satisfies an internal boundary condition. Curiously, most computer codes implementing adjoint error estimation ignore the new term in the functional, as well as the internal adjoint boundary condition. The purpose of this note is to justify this omission as follows: if one represents the exact forward and adjoint solutions as vanishing viscosity limits of the corresponding viscous problems, then the internal boundary condition is naturally satisfied in the limit.
KEYWORDS	adjoint error control, conservation laws, discontinuities, interior boundary condition
DOI	10.1137/120870335
PUBLICATION	SIAM Journal on Numerical Analysis 51(1), 126–136. (11 pages)