## Error analysis

In 1996, Endre Suli and I realised the importance of the adjoint solution in analysing the numerical error in integral functionals, such as lift and drag, which are often the quantities of most interest in engineering computations. This led to a report on the error analysis for finite element approximations of the incompressible Navier-Stokes equations [1] in which we established an important superconvergence property, that the order of accuracy of the functional increases twice as quickly as the order of the polynomial finite element function space.

In 1998, Niles Pierce and I wrote a paper in which we developed a closed form solution for the adjoint quasi-1D Euler equations, with and without shocks, and showed excellent agreement with numerical computations [2]. In this paper we also showed how the error in the `lift' (the integral of pressure along the quasi-1D duct) could be improved by correcting the leading order effects of the truncation error in the finite volume method; I had outlined this idea the previous year in a short paper for a conference honouring Earll Murman [3].

We then realised that the error correction was much more easily constructed by interpolating the numerical finite volume solution to obtain an approximate analytic solution, and then using this to evaluate the residual error and thereby form an inner product with an approximate adjoint solution to correct the computed value for the functional. This has been the central idea behind a sequence of papers [4-8] in which we have developed the analytic theory and demonstrated superconvergent correct functionals for a variety of problems, in each case using the adjoint correction to double the the order of accuracy of output functionals. Our most comprehensive reference on the topic is a set of lectures notes prepared for a course which was given at both NASA Ames and VKI [9], while our latest results for the 2D Euler equations will appear in [12].

In 2002, Endre Suli and I wrote a lengthy Acta Numerica paper [10] on his research on a posteriori error analysis and optimal grid adaptation, and my work on adjoint error correction.

My latest research in this area has been on adjoint error correction when there is a shock in the underlying nonlinear solution. Here again it has been possible to obtain fourth order functional accuracy from a solution which is only second order accurate [11,12].

## References

1. M.B. Giles, M.G. Larson, J.M. Levenstam and E. Suli. `Adaptive error control for finite element approximations of the lift and drag coefficients in viscous flow'. Report NA-97/06, Oxford University Computing Laboratory, 1997.
2. M.B. Giles and N.A. Pierce. `On the properties of solutions of the adjoint Euler equations'. 6th ICFD Conference on Numerical Methods for Fluid Dynamics, Oxford, UK, 1998. (PDF)
3. M.B. Giles. `On adjoint equations for error analysis and optimal grid adaptation', in Frontiers of Computational Fluid Dynamics 1998, 155-170. D.A. Caughey and M.M. Hafez editors, World Scientific, 1998. Report NA-97/11, Oxford University Computing Laboratory, 1997.
4. M.B. Giles and N.A. Pierce `Adjoint recovery of superconvergent functionals from approximate solutions of partial differential equations'. Report NA-98/18. Oxford University Computing Laboratory, 1998.
5. M.B. Giles and N.A. Pierce. `Improved lift and drag estimates using adjoint Euler equations'. AIAA Paper 99-3293, 1999. (PDF)
6. N.A. Pierce and M.B. Giles. `Adjoint recovery of superconvergent functionals from PDE approximations'. SIAM Review, 42(2):247-264, 2000. (PDF)
7. M.B. Giles and N.A. Pierce. `Superconvergent lift estimates through adjoint error analysis', in Innovative Methods for Numerical Solutions of Partial Differential Equations. M.M. Hafez and J.J. Chattot editors, World Scientific, 2001. (PDF)
8. M.B. Giles. `Defect and adjoint error correction', in Computational Fluid Dynamics 2000. N. Satofuka, editor, Springer-Verlag, 2001. (PDF).
9. M.B. Giles and N.A. Pierce. `Adjoint error correction for integral outputs', in Error Estimation and Adaptive Discretization Methods in Computational Fluid Dynamics, pages 47-96, editors T. Barth and H. Deconinck, volume 25 in Lecture Notes in Computational Science and Engineering. Springer-Verlag, 2002. (PDF).
10. M.B. Giles and E. Suli. `Adjoint methods for PDEs: a posteriori error analysis and postprocessing by duality'. Acta Numerica 2002, pages 145-236, Cambridge University Press, 2002. (PDF file: 2.1Mb).
11. M.B. Giles, N.A. Pierce and E. Suli . `Progress in adjoint error correction for integral functionals', Computing and Visualisation in Science, 6(2-3), 2004. (PDF file: 3.3Mb).
12. N.A. Pierce and M.B. Giles. `Adjoint and Defect Error Bounding and Correction for Functional Estimates' Journal of Computational Physics, 200:769-794, 2004. (PDF file: 426kb).