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

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.

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

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

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

[4]  W. K. Anderson,et al.  Aerodynamic design optimization on unstructured grids with a continuous adjoint formulation , 1997 .

[5]  A. Jameson,et al.  Optimum Aerodynamic Design Using the Navier–Stokes Equations , 1997 .

[6]  Antony Jameson,et al.  Aerodynamic design via control theory , 1988, J. Sci. Comput..

[7]  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 .

[8]  A. Jameson,et al.  Aerodynamic Shape Optimization of Complex Aircraft Configurations via an Adjoint Formulation , 1996 .

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

[10]  Jaime Peraire,et al.  Practical 3D aerodynamic design and optimization using unstructured meshes , 1996 .

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

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

[13]  M. Giles.  On adjoint equations for error analysis and optimal grid adaptation in CFD , 1997 .

[14]  H. Keller,et al.  Approximation methods for nonlinear problems with application to two-point boundary value problems , 1975 .

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

[16]  Michael B. Giles,et al.  Analysis of the accuracy of shock-capturing in the steady quasi-1D Euler equations , 1995 .

[17]  M. Giles,et al.  Adjoint equations in CFD: duality, boundary conditions and solution behaviour , 1997 .

[18]  J. M. Sanz-Serna,et al.  Stability and Convergence in Numerical Analysis III: Linear Investigation of Nonlinear Stability , 1988 .

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