Adjoint Recovery of Superconvergent Functionals from PDE Approximations

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.

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

[2]  K. Bohmer.  Defect Correction Methods: Theory and Applications , 1984 .

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

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

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

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

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

[8]  L. Richardson,et al.  The Deferred Approach to the Limit. Part I. Single Lattice. Part II. Interpenetrating Lattices , 1927 .

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

[10]  Michael B. Giles,et al.  Analytic Adjoint Solutions for the Quasi-1D Euler Equations , 2000 .

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

[12]  Klaus Böhmer,et al.  Defect Correction Methods , 1984, Computing Supplementum.

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

[14]  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 .

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

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

[17]  A. Booth.  Numerical Methods , 1957, Nature.

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

[19]  Heinz-Otto Kreiss,et al.  Difference approximations for boundary and eigenvalue problems for ordinary differential equations , 1972 .

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

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

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

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

[24]  A. Brandt.  Guide to multigrid development , 1982 .

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

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

[27]  M. J. Rimlinger,et al.  Constrained Multipoint Aerodynamic Shape Optimization Using an Adjoint Formulation and Parallel Computers , 1997 .

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

[29]  Åke Björck,et al.  Numerical Methods , 1995, Handbook of Marine Craft Hydrodynamics and Motion Control.

[30]  L. Fox,et al.  Some improvements in the use of relaxation methods for the solution of ordinary and partial differential equations , 1947, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

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