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Adjoint recovery of superconvergent functionals from PDE approximations

Pierce, Niles A. and Giles, Michael B. (2000) Adjoint recovery of superconvergent functionals from PDE approximations. SIAM Review, 42 (2). pp. 247-264. ISSN 0036-1445. doi:10.1137/S0036144598349423.

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Motivated by applications in computational fluid dynamics, a method is presented 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 that uses an adjoint PDE 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 approximate solution to the adjoint equations may thus be combined to produce a correction for the computed functional value that yields the desired improvement in accuracy. Numerical results are presented for the Poisson equation in one and two dimensions and for the nonlinear quasi-one-dimensional Euler equations. The theory is equally applicable to nonlinear equations in complex multi-dimensional domains and holds great promise for use in a range of engineering disciplines in which a few integral quantities are a key output of numerical approximations.

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Pierce, Niles A.0000-0003-2367-4406
Additional Information:© 2000 Society for Industrial and Applied Mathematics. Received by the editors December 17, 1998; accepted for publication (in revised form) September 12, 1999; published electronically April 24, 2000. This work was supported by EPSRC grant GR/K91149. During this research, we have benefited from many discussions with our colleagues Dr. Endre Suli and Dr. Paul Houston, who are also researching adjoint error analysis and optimal grid adaptation for finite element methods.
Funding AgencyGrant Number
Engineering and Physical Sciences Research Council (EPSRC)GR/K91149
Subject Keywords:PDEs, adjoint equations, error analysis, superconvergence, post-processing approach, finite-element method, aerodynamic design, differential equations, optimization, formulation, boundary
Issue or Number:2
Record Number:CaltechAUTHORS:PIEsiamrev00
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:808
Deposited By: Tony Diaz
Deposited On:06 Oct 2005
Last Modified:08 Nov 2021 19:05

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