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A posteriori error bounds for reduced-basis approximation of parametrized noncoercive and nonlinear elliptic partial differential equations

Patera, Adolf, 1836-1912, Rovas Dimitrios, C. Prud’homme, Veroy, Karen

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Summary

We present a technique for the rapid and reliable prediction of linear–functional out- puts of elliptic partial differential equations with affine parameter dependence. The essential components are (i) rapidly convergent global reduced–basis approximations — (Galerkin) projection onto a space WN spanned by solutions of the governing partial dif- ferential equation at N selected points in parameter space; (ii) a posteriori error estimation — relaxations of the error-residual equation that provide inexpensive yet sharp bounds for the error in the outputs of interest; and (iii) off-line/on-line computational procedures — methods which decouple the generation and projection stages of the approximation process. The operation count for the on–line stage — in which, given a new parameter value, we calculate the output of interest and associated error bound — depends only on N (typically very small) and the parametric complexity of the problem.In this paper we develop new a posteriori error estimation procedures for noncoercive linear, and certain nonlinear, problems that yield rigorous and sharp error statements for all N. We consider three particular examples: the Helmholtz (reduced-wave) equation; a cubically nonlinear Poisson equation; and Burgers equation — a model for incompressible Navier-Stokes. The Helmholtz (and Burgers) example introduce our new lower bound constructions for the requisite inf-sup (singular value) stability factor; the cubic nonlin- earity exercises symmetry factorization procedures necessary for treatment of high-order Galerkin summations in the (say) residual dual-norm calculation; and the Burgers equa- tion illustrates our accommodation of potentially multiple solution branches in our a posteriori error statement. Numerical results are presented that demonstrate the rigor, sharpness, and efficiency of our proposed error bounds, and the application of these bounds to adaptive (optimal) approximation.

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