A hierarchical duality approach to bounds for the outputs of partial differential equations. (English) Zbl 0953.76054

Summary: We present a technique for generating lower and upper bounds on outputs which are linear functionals of exact or finite element solutions to symmetric or nonsymmetric coercive linear partial differential equations. The method is based on the construction of an augmented Lagrangian which integrates a quadratic ‘energy’ reformulation of the desired output as the objective to be minimized, with finite-element equilibrium equations and (conforming) ‘hybridized’ intersubdomain continuity conditions. The bounds are then derived by appealing to the associated dual unconstrained max-min problem evaluated for optimally chosen Lagrange multipliers generated by a less expensive approximation, such as a low-dimensional finite-element discretization. As in many a posteriori error estimation techniques, the bound calculation requires only the solution of subdomain-local symmetric (Neumann) problems on the refined ‘true’ mesh. The technique is illustrated in the case of one-dimensional convection-diffusion equation.


76M10 Finite element methods applied to problems in fluid mechanics
76R99 Diffusion and convection
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
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[1] Ainsworth, M.; Oden, J.T., A unified approach to a posteriori error estimation using element residual methods, Numer. math., 65, 23-50, (1993) · Zbl 0797.65080
[2] Ainsworth, M.; Oden, J.T., A posteriori error estimation in finite element analysis, TICAM report 96-19, (1996)
[3] N. Alexandrov, J.E. Dennis, Jr., R. M. Lewis and V. Torczon, A trust region framework for managing the use of approximation models in optimization, in preparation.
[4] Anagnostou, G.; Maday, Y.; Mavriplis, C.; Patera, A.T., On the mortar element method: generalizations and implementation, (), 157-173
[5] Azaiez, M.; Bernardi, C.; Maday, Y., Some tools for adaptivity in the spectral element method, ()
[6] Bank, R., Analysis of a local a posteriori error estimates for elliptic equations, (), 119-128
[7] Bank, R.E.; Weiser, A., Some a posteriori error estimators for elliptic partial differential equations, Math. comput., 44, 170, 283-301, (1985) · Zbl 0569.65079
[8] Bar-Yoseph, P.; Israeli, M., An asymptotic finite element method for improvement of solutions of boundary layer problems, Numer. math., 49, 425-438, (1986) · Zbl 0616.65105
[9] Becker, R.; Rannacher, R., Weighted a posteriori error control in finite element methods, IWR preprint 96-1 (SFB 359), (1996), Heidelberg · Zbl 0868.65076
[10] Bernardi, C.; Maday, Y.; Patera, A.T., A new nonconforming approach to domain decomposition: the mortar element method, () · Zbl 0797.65094
[11] Brezzi, F.; Fortin, M., Mixed and hybrid finite element methods, (1991), Springer-Verlag New York · Zbl 0788.73002
[12] Colmar, V., A proper orthogonal decomposition approximation and bound technique, (), and M.I.T.
[13] Deville, M.O.; Mund, E., Chebyshev pseudospectral solution of second-order elliptic equations with finite element preconditioning, J. comput. phys., 60, 517-533, (1985) · Zbl 0585.65073
[14] Eriksson, K.; Johnson, C., An adaptive finite element method for linear elliptic problems, Math. comput., 50, 361-383, (1988) · Zbl 0644.65080
[15] Glowinski, R.; Tallec, P.Le, Augmented Lagrangian and operator-splitting methods in nonlinear mechanics, (1989), SIAM Philadelphia · Zbl 0698.73001
[16] Kikuchi, N., Finite element methods in mechanics, (1986), Cambridge University Press Cambridge · Zbl 0587.73102
[17] Fischer, P.F.; Patera, A.T., Parallel simulation of viscous incompressible flows, Ann. rev. fluid mech., 26, 483-527, (1994) · Zbl 0802.76065
[18] Ladeveze, P.; Leguillon, D., Error estimation procedures in the finite element method and applications, SIAM J. numer. anal., 20, 485-509, (1983) · Zbl 0582.65078
[19] Maday, Y.; Meiron, D.I.; Patera, A.T.; Rønquist, E.M., Analysis of iterative methods for the steady and unsteady Stokes problem: application to spectral element discretizations, SIAM J. sci. comput., 14, 310-337, (1993) · Zbl 0769.76047
[20] Orszag, S.A., Spectral methods for problems in complex geometries, J. comput. phys., 37, 70-92, (1980) · Zbl 0476.65078
[21] Paraschivoiu, M., A posteriori finite element bounds for linear-functional outputs of coercive partial differential equations and of the Stokes problem, ()
[22] M. Paraschivoiu, J. Peraire and A. T. Patera, A posteriori finite element bounds for linear-functional outputs of elliptic partial differential equations, Comput. Methods Appl. Mech. Engrg., to appear. · Zbl 0907.65102
[23] J. Peraire and A. T. Patera, A posteriori finite element bounds for outputs of nonlinear boundary value problems, in preparation. · Zbl 0943.65124
[24] Strang, G., Introduction to applied mathematics, (1986), Wellesley-Cambridge Press Wellesley, Massachusetts · Zbl 0618.00015
[25] Strang, G.; Fix, G.J., An analysis of the finite element method, (1973), Prentice-Hall Englewood Cliffs, New Jersey · Zbl 0278.65116
[26] Vailong, H., A posteriori bounds for linear-functional outputs of hyperbolic partial differential equations, ()
[27] Verfüth, R., A posteriori error estimation and adaptive mesh-refinement techniques, J. comput. appl. math., 50, 67-83, (1994) · Zbl 0811.65089
[28] Yeşilyurt, S.; Paraschivoiu, M.; Otto, J.; Patera, A.T., Computer-simulation response surfaces: a Bayesian-validated approach, (), 13-22
[29] Yeşilyurt, S.; Patera, A.T., Surrogates for numerical simulations; optimization of eddy-promoter heat exchangers, Comput. methods appl. mech. engrg., 121, 231-257, (1995) · Zbl 0852.76074
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