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8 - Discretization error

from Part III - Solution verification

Published online by Cambridge University Press:  05 March 2013

Christopher J. Roy
Affiliation:
Virginia Polytechnic Institute and State University
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Publisher: Cambridge University Press
Print publication year: 2010

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References

Ainsworth, M. and Oden, J. T. (1997). A posteriori error estimation in finite element analysis, Computer Methods in Applied Mechanics and Engineering. 142(1–2), 1–88.CrossRefGoogle Scholar
Ainsworth, M. and Oden, J. T. (2000). A Posteriori Error Estimation in Finite Element Analysis, New York, Wiley Interscience.CrossRefGoogle Scholar
Akin, J. E. (2005). Finite Element Analysis with Error Estimators, Burlington, Elsevier.Google Scholar
Babuska, I. and Miller, A. (1984). Post-processing approach in the finite element method – Part 3: A posteriori error estimates and adaptive mesh selection, International Journal for Numerical Methods in Engineering. 20(12), 2311–2324.CrossRefGoogle Scholar
Babuska, I. and Rheinboldt, W. C. (1978a). A posteriori error estimates for the finite element method, International Journal for Numerical Methods in Engineering. 12, 1597–1615.CrossRefGoogle Scholar
Babuska, I. and Rheinboldt, W. C. (1978b). Error estimates for adaptive finite element computations, SIAM Journal of Numerical Analysis. 15(4), 736–754.CrossRefGoogle Scholar
Babuska, I., Zienkiewicz, O. C., Gago, J., Oliveira, E. R. (1986). Accuracy Estimates and Adaptive Refinements in Finite Element Computations, Chichester, Wiley.Google Scholar
Babuska, I., Strouboulis, T., and Upadhyay, C. S. (1994). A model study of the quality of a posteriori error estimators for linear elliptic problems. Error estimation in the interior of patchwise uniform grids of triangles, Computer Methods in Applied Mechanics and Engineering. 114(3–4), 307–378.CrossRefGoogle Scholar
Babuska, I., Strouboulis, T., Gangaraj, T., and Upadhyay, C. S. (1997). Pollution error in the h-version of the finite element method and local quality of the recovered derivative, Computer Methods in Applied Mechanics and Engineering. 140(1–2), 1–37.CrossRefGoogle Scholar
Baker, T. J. (2005). On the Relationship between Mesh Refinement and Solution Accuracy, AIAA Paper 2005–4875.
Bank, R. E. (1996). Hierarchical bases and the finite element method, Acta Numerica. 5, 1–45.CrossRefGoogle Scholar
Bank, R. R. and Weiser, A. (1985). Some a posteriori error estimators for elliptic partial differential equations, Mathematics of Computation. 44, 283–301.CrossRefGoogle Scholar
Banks, J. W., Aslam, T., and Rider, W. J. (2008). On sub-linear convergence for linearly degenerate waves in capturing schemes, Journal of Computational Physics. 227, 6985–7002.CrossRefGoogle Scholar
Barth, T. J. and Larson, M. G. (2002). A-posteriori error estimation for higher order Godunov finite volume methods on unstructured meshes, In Finite Volumes for Complex Applications III, Herbin, R. and Kroner, D. (eds.), London, HERMES Science Publishing Ltd., 41–63.Google Scholar
Brock, J. S. (2007). Bounded Numerical Error Estimates for Oscillatory Convergence of Simulation Data, AIAA Paper 2007–4091.
Cadafalch, J., Perez-Segarra, C. D., Consul, R., and Oliva, A. (2002). Verification of finite volume computations on steady-state fluid flow and heat transfer, Journal of Fluids Engineering. 24, 11–21.CrossRefGoogle Scholar
Carpenter, M. H. and Casper, J. H. (1999). Accuracy of shock capturing in two spatial dimensions, AIAA Journal. 37(9), 1072–1079.CrossRefGoogle Scholar
Cavallo, P. A. and Sinha, N. (2007). Error quantification for computational aerodynamics using an error transport equation, Journal of Aircraft. 44(6), 1954–1963.CrossRefGoogle Scholar
Celik, I., Li, J., Hu, G., and Shaffer, C. (2005). Limitations of Richardson extrapolation and some possible remedies, Journal of Fluids Engineering. 127, 795–805.CrossRefGoogle Scholar
Cheng, Z. and Paraschivoiu, M. (2004). A posteriori finite element bounds to linear functional outputs of the three-dimensional Navier–Stokes equations, International Journal for Numerical Methods in Engineering. 61(11), 1835–1859.CrossRefGoogle Scholar
Coleman, H. W., Stern, F., Di Mascio, A., and Campana, E. (2001). The problem with oscillatory behavior in grid convergence studies, Journal of Fluids Engineering. 123, 438–439.CrossRefGoogle Scholar
Demkowicz, L., Oden, J. T., and Strouboulis, T. (1984). Adaptive finite elements for flow problems with moving boundaries. Part I: Variational principles and a posteriori estimates, Computer Methods in Applied Mechanics and Engineering. 46(2), 217–251.CrossRefGoogle Scholar
Eca, L. and Hoekstra, M. (2002). An evaluation of verification procedures for CFD applications, 24th Symposium on Naval Hydrodynamics, Fukuoka, Japan, July 8–13, 2002.
Eca, L. and Hoekstra, M. (2009a). Error estimation based on grid refinement studies: a challenge for grid generation, Congress on Numerical Methods in Engineering, Barcelona, Spain, June 29–July 2, 2009.
Eca, L. and Hoekstra, M. (2009b). Evaluation of numerical error estimation based on grid refinement studies with the method of the manufactured solutions, Computers and Fluids. 38, 1580–1591.CrossRefGoogle Scholar
Eca, L., Vaz, G. B., Falcao de Campos, J. A. C., and Hoekstra, M. (2004). Verification of calculations of the potential flow around two-dimensional foils, AIAA Journal. 42(12), 2401–2407.CrossRefGoogle Scholar
Eca, L., Hoekstra, M., and Roache, P. (2005). Verification of Calculations: an Overview of the Lisbon Workshop, AIAA Paper 2005–4728.
Eriksson, K. and Johnson, C. (1987). Error-estimates and automatic time step control for nonlinear parabolic problems. Part 1, SIAM Journal of Numerical Analysis. 24(1), 12–23.CrossRefGoogle Scholar
Estep, D., Larson, M., and Williams, R. (2000). Estimating the Error of Numerical Solutions of Systems of Nonlinear Reaction-Diffusion Equations, Memoirs of the American Mathematical Society, Vol. 146, No. 696, Providence, American Mathematical Society.Google Scholar
Fehlberg, E. (1969). Low-Order Classical Runge-Kutta Formulas with Step Size Control and their Application to some Heat Transfer Problems, NASA Technical Report 315, National Aeronautics and Space Administration, July 1969.
Ferziger, J. H. and Peric, M. (1996). Further discussion of numerical errors in CFD, International Journal for Numerical Methods in Fluids. 23(12), 1263–1274.3.0.CO;2-V>CrossRefGoogle Scholar
Ferziger, J. H. and Peric, M. (2002). Computational Methods for Fluid Dynamics, 3rd edn., Berlin, Springer-Verlag.CrossRefGoogle Scholar
Garbey, M. and Shyy, W. (2003). A least square extrapolation method for improving solution accuracy of PDE computations, Journal of Computational Physics. 186(1), 1–23.CrossRefGoogle Scholar
Hirsch, C. (1990). Numerical Computation of Internal and External Flows: Volume 2, Computational Methods for Inviscid and Viscous Flows, Chichester, Wiley.Google Scholar
Hirsch, C. (2007). Numerical Computation of Internal and External Flows: the Fundamentals of Computational Fluid Dynamics, 2nd edn., Oxford, Butterworth-Heinemann.Google Scholar
Huebner, K. H. (2001). The Finite Element Method for Engineers, New York, Wiley.Google Scholar
Huebner, K. H., Dewhirst, D. L., Smith, D. E., and Byrom, T. G. (2001). The Finite Element Method of Engineers, 4th edn., New York, John Wiley and Sons.Google Scholar
Hughes, T. J. R. (2000). The Finite Element Method: Linear Static and Dynamic Finite Element Analysis, 2nd edn., Mineola, Dover.Google Scholar
Jameson, A. (1988). Aerodynamic design via control theory, Journal of Scientific Computing. 3(3), 233–260.CrossRefGoogle Scholar
Jameson, A., Schmidt, W., and Turkel, E. (1981). Numerical Solutions of the Euler Equations by Finite Volume Methods Using Runge-Kutta Time-Stepping Schemes, AIAA Paper 81–1259.
Johnson, C. and Hansbo, P. (1992). Adaptive finite element methods in computational mechanics, Computer Methods in Applied Mechanics and Engineering. 101(1–3), 143–181.CrossRefGoogle Scholar
Kamm, J. R., Rider, W. J., and Brock, J. S. (2003). Combined Space and Time Convergence Analyses of a Compressible Flow Algorithm, AIAA Paper 2003–4241.
King, M. L., Fisher, M. J., and Jensen, C. G. (2006). A CAD-centric approach to CFD analysis with discrete features, Computer-Aided Design & Applications. 3(1–4), 279–288.CrossRefGoogle Scholar
Knight, D. D. (2006). Elements of Numerical Methods for Compressible Flows, New York, Cambridge University Press.CrossRefGoogle Scholar
Moin, P. (2007). Application of high fidelity numerical simulations for vehicle aerodynamics, The Aerodynamics of Heavy Vehicles II: Trucks, Buses and Trains, Tahoe City, California, August 26–31, 2007.Google Scholar
Morton, K. W. and Mayers, D. F. (2005). Numerical Solution of Partial Differential Equations: an Introduction, 2nd edn., New York, Cambridge University Press.CrossRefGoogle Scholar
Oden, J. T. and Reddy, J. N. (1976). An Introduction to the Mathematical Theory of Finite Elements, New York, Wiley.Google Scholar
Paraschivoiu, M., Peraire, J., and Patera, A. T. (1997). A posteriori finite element bounds for linear functional outputs of elliptic partial differential equations, Computer Methods in Applied Mechanics and Engineering. 150(1–4), 289–312.CrossRefGoogle Scholar
Pelletier, D. and Roache, P. J. (2006). Chapter 13: Verification and validation of computational heat transfer, in Handbook of Numerical Heat Transfer, 2nd edn., W. J. Minkowycz, E. M. Sparrow, and J. Y. Murthy, eds., Hoboken, NJ, Wiley.Google Scholar
Pierce, N. A. and Giles, M. B. (2000). Adjoint recovery of superconvergent functionals from PDE approximations, SIAM Review. 42(2), 247–264.CrossRefGoogle Scholar
Potter, D. L., Blottner, F. G., Black, A. R., Roy, C. J., and Bainbridge, B. L. (2005). Visualization of Instrumental Verification Information Details (VIVID): Code Development, Description, and Usage, SAND2005–1485, Albuquerque, NM, Sandia National Laboratories.
Rannacher, R. and Suttmeier, F. T. (1997). A feed-back approach to error control in finite element methods: application to linear elasticity, Computational Mechanics. 19(5), 434–446.CrossRefGoogle Scholar
Richards, S. A. (1997). Completed Richardson extrapolation in space and time, Communications in Numerical Methods in Engineering. 13, 1997, 573–582.3.0.CO;2-6>CrossRefGoogle Scholar
Richardson, L. F. (1911). The approximate arithmetical solution by finite differences of physical problems involving differential equations, with an application to the stresses in a masonry dam, Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character. 210, 307–357.CrossRefGoogle Scholar
Richardson, L. F. (1927). The deferred approach to the limit. Part I. Single lattice, Philosophical Transaction of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character. 226, 299–349.CrossRefGoogle Scholar
Richtmyer, R. and Morton, K. (1967). Difference Methods for Initial-Value Problems, 2nd edn., New York, Interscience Publishers.Google Scholar
Roache, P. J. (1994). Perspective: a method for uniform reporting of grid refinement studies, Journal of Fluids Engineering. 116, 405–413.CrossRefGoogle Scholar
Roache, P. J. (1998). Verification and Validation in Computational Science and Engineering, Albuquerque, NM, Hermosa Publishers.Google Scholar
Roache, P. J. (2003a). Conservatism of the grid convergence index in finite volume computations on steady-state fluid flow and heat transfer, Journal of Fluids Engineering. 125(4), 731–732.CrossRefGoogle Scholar
Roache, P. J. (2003b). Criticisms of the ‘‘correction factor’’ verification method, Journal of Fluids Engineering. 125(4), 732–733.CrossRefGoogle Scholar
Roache, P. J. (2009). Private communication, July 13, 2009.
Roache, P. J. and Knupp, P. M. (1993). Completed Richardson extrapolation, Communications in Numerical Methods in Engineering. 9(5), 365–374.CrossRefGoogle Scholar
Roy, C. J. (2001). Grid Convergence Error Analysis for Mixed-Order Numerical Schemes, AIAA Paper 2001–2606.
Roy, C. J. (2003). Grid convergence error analysis for mixed-order numerical schemes, AIAA Journal. 41(4), 595–604.CrossRefGoogle Scholar
Roy, C. J. (2005). Review of code and solution verification procedures for computational simulation, Journal of Computational Physics. 205(1), 131–156.CrossRefGoogle Scholar
Roy, C. J. (2009). Strategies for Driving Mesh Adaptation in CFD, AIAA Paper 2009–1302.
Roy, C. J. (2010). Review of Discretization Error Estimators in Scientific Computing, AIAA Paper 2010–126.
Roy, C. J. and Blottner, F. G. (2003). Methodology for turbulence model validation: application to hypersonic transitional flows, Journal of Spacecraft and Rockets. 40(3), 313–325.CrossRefGoogle Scholar
Roy, C. J., Heintzelman, C. J., and Roberts, S. J. (2007). Estimation of Numerical Error for 3D Inviscid Flows on Cartesian Grids, AIAA Paper 2007–0102.
Salas, M. D. (2006). Some observations on grid convergence, Computers and Fluids. 35, 688–692.CrossRefGoogle Scholar
Salas, M. D. and Atkins, H. L. (2009). On problems associated with grid convergence of functionals, Computers and Fluids. 38, 1445–1454.CrossRefGoogle Scholar
Shih, T. I.-P. and Qin, Y. C. (2007). A Posteriori Method for Estimating and Correcting Grid-Induced Errors in CFD Solutions Part 1: Theory and Method, AIAA Paper 2007–100.
Shih, T. I.-P., and Williams, B. R. (2009). Development and Evaluation of an A Posteriori Method for Estimating and Correcting Grid-Induced Errors in Solutions of the Navier-Stokes Equations, AIAA Paper 2009–1499.
Sonar, T. (1993). Strong and weak norm refinement indicators based on the finite element residual for compressible flow computation: I. The steady case, Impact of Computing in Science and Engineering. 5(2), 111–127.CrossRefGoogle Scholar
Stewart, J. R. and Hughes, T. J. R. (1998). A tutorial in elementary finite element error analysis: a systematic presentation of a priori and a posteriori error estimates, Computer Methods in Applied Mechanics and Engineering. 158(1–2), 1–22.CrossRefGoogle Scholar
Stern, F., Wilson, R. V., Coleman, H. W., and Paterson, E. G. (2001). Comprehensive approach to verification and validation of CFD simulations – Part I: Methodology and procedures, ASME Journal of Fluids Engineering. 123(4), 793–802.CrossRefGoogle Scholar
Szabo, B. A. and Babuska, I. (1991). Finite Element Analysis, New York, Wiley.Google Scholar
Tannehill, J. C., Anderson, D. A., and Pletcher, R. H. (1997). Computational Fluid Mechanics and Heat Transfer, 2nd edn., Philadelphia, Taylor and Francis.Google Scholar
Thompson, J. F., Warsi, Z. U. A., and Mastin, C. W. (1985). Numerical Grid Generation: Foundations and Applications, New York, Elsevier. (www.erc.msstate.edu/publications/gridbook)Google Scholar
Venditti, D. A. and Darmofal, D. L. (2000). Adjoint error estimation and grid adaptation for functional outputs: application to quasi-one dimensional flow, Journal of Computational Physics. 164, 204–227.CrossRefGoogle Scholar
Venditti, D. A. and Darmofal, D. L. (2002). Grid adaptation for functional outputs: application to two-dimensional inviscid flows, Journal of Computational Physics. 176, 40–69.CrossRefGoogle Scholar
Venditti, D. A. and Darmofal, D. L. (2003). Anisotropic grid adaptation for functional outputs: application to two-dimensional viscous flows, Journal of Computational Physics. 187, 22–46.CrossRefGoogle Scholar
Wahlbin, L. B. (1995). Superconvergence in Galerkin Finite Element Methods, Volume 1605 of Lecture Notes in Mathematics, Springer-Verlag, Berlin.CrossRefGoogle Scholar
Whiteman, J. R. (1994). The Mathematics of Finite Element and Applications: Highlights 1993, New York, Wiley.Google Scholar
Xing, T. and Stern, F. (2009). Factors of Safety for Richardson Extrapolation, IIHR Hydroscience and Engineering Technical Report No. 469, March 2009.
Zhang, X. D., Trepanier, J.-Y., and Camarero, R. (2000). A posteriori error estimation for finite-volume solutions of hyperbolic conservation laws, Computer Methods in Applied Mechanics and Engineering. 185(1), 1–19.CrossRefGoogle Scholar
Zhang, Z. and Naga, A. (2005). A new finite element gradient recovery method: superconvergence property, SIAM Journal of Scientific Computing. 26(4), 1192–1213.CrossRefGoogle Scholar
Zienkiewicz, O. C., Taylor, R. L., and Zhu, J. Z. (2005). The Finite Element Method : Its Basis and Fundamentals, 6th edn., Oxford, Elsevier.Google Scholar
Zienkiewicz, O. C. and Zhu, J. Z. (1987). A simple error estimator and adaptive procedure for practical engineering analysis, International Journal for Numerical Methods in Engineering. 24, 337–357.CrossRefGoogle Scholar
Zienkiewicz, O. C. and Zhu, J. Z. (1992). The superconvergent patch recovery and a posteriori error estimates, Part 2: Error estimates and adaptivity, International Journal for Numerical Methods in Engineering. 33, 1365–1382.CrossRefGoogle Scholar

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