Progress in adjoint error correction for integral functionals

When approximating the solutions of partial differential equations, it is a few key output integrals which are often of most concern. This paper shows how the accuracy of these values can be improved through a correction term which is an inner product of the residual error in the original p.d.e. and the solution of an appropriately defined adjoint p.d.e. A number of applications are presented and the challenges of smooth reconstruction on unstructured grids and error correction for shocks are discussed.

[1]  G. Strang.  VARIATIONAL CRIMES IN THE FINITE ELEMENT METHOD , 1972 .

[2]  J. Craggs.  Applied Mathematical Sciences , 1973 .

[3]  Ivo Babuška,et al.  The post-processing approach in the finite element method—part 1: Calculation of displacements, stresses and other higher derivatives of the displacements , 1984 .

[4]  Ivo Babuška,et al.  The post‐processing approach in the finite element method—Part 2: The calculation of stress intensity factors , 1984 .

[5]  John W. Barrett,et al.  Total Flux Estimates for a Finite-Element Approximation of Elliptic Equations , 1986 .

[6]  R. Kress,et al.  Inverse Acoustic and Electromagnetic Scattering Theory , 1992 .

[7]  Arthur Veldman,et al.  NUMERICAL METHODS FOR FLUID DYNAMICS 4 , 1993 .

[8]  Claes Johnson,et al.  Numerics and hydrodynamic stability: toward error control in computational fluid dynamics , 1995 .

[9]  Endre Süli,et al.  Finite element methods for hyperbolic problems: a posteriori error analysis and adaptivity , 1996 .

[10]  Endre Süli,et al.  A Posteriori Error Analysis And Adaptivity For Finite Element Approximations Of Hyperbolic Problems , 1997 .

[11]  J. Peraire,et al.  A posteriori finite element bounds for linear-functional outputs of elliptic partial differential equations , 1997 .

[12]  Richard G. Compton,et al.  A General Method for Electrochemical Simulations. 1. Formulation of the Strategy for Two-Dimensional Simulations , 1997 .

[13]  Anthony T. Patera,et al.  Bounds for Linear–Functional Outputs of Coercive Partial Differential Equations : Local Indicators and Adaptive Refinement , 1998 .

[14]  J. Tinsley Oden,et al.  Advances in adaptive computational methods in mechanics , 1998 .

[15]  Endre Süli,et al.  The Adaptive Computation of Far-Field Patterns by A Posteriori Error Estimation of Linear Functionals , 1998 .

[16]  Michael B. Giles,et al.  Adjoint recovery of superconvergent functionals from approximate solutions of partial differential equations , 1998 .

[17]  Christian Rohde,et al.  An Introduction to Recent Developments in Theory and Numerics for Conservation Laws: Proceedings of the International School on Theory and Numerics for Conservation Laws, Freiburg/Littenweiler, Germany, October 20-24, 1997 , 1999, Theory and Numerics for Conservation Laws.

[18]  Michael B. Giles,et al.  Improved- lift and drag estimates using adjoint Euler equations , 1999 .

[19]  Michael B. Giles,et al.  Adjoint Recovery of Superconvergent Functionals from PDE Approximations , 2000, SIAM Rev..

[20]  D. Venditti,et al.  Adjoint error estimation and grid adaptation for functional outputs: application to quasi-one-dimensional flow , 2000 .

[21]  Michael B. Giles,et al.  Analytic adjoint solutions for the quasi-one-dimensional Euler equations , 2001, Journal of Fluid Mechanics.

[22]  Stefan Ulbrich,et al.  A Sensitivity and Adjoint Calculus for Discontinuous Solutions of Hyperbolic Conservation Laws with Source Terms , 2002, SIAM J. Control. Optim..

[23]  D. Venditti,et al.  Grid adaptation for functional outputs: application to two-dimensional inviscid flows , 2002 .

[24]  Endre Süli,et al.  Acta Numerica 2002: Adjoint methods for PDEs: a posteriori error analysis and postprocessing by duality , 2002 .

[25]  M. Giles,et al.  Adjoint Error Correction for Integral Outputs , 2003 .

[26]  Michael B. Giles,et al.  Discrete Adjoint Approximations with Shocks , 2003 .

[27]  Stefan Ulbrich,et al.  Adjoint-based derivative computations for the optimal control of discontinuous solutions of hyperbolic conservation laws , 2003, Syst. Control. Lett..

[28]  E. Tadmor,et al.  Hyperbolic Problems: Theory, Numerics, Applications , 2003 .

[29]  T. Barth,et al.  Error estimation and adaptive discretization methods in computational fluid dynamics , 2003 .

[30]  Arieh Iserles,et al.  Acta Numerica 2004 , 2004 .