A Posteriori Error Bounds for Reduced-Basis Approximation of Parametrized Noncoercive and Nonlinear Elliptic Partial Differential Equations

Abstract

We present a technique for the rapid and reliable prediction of linear-functional outputs 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 W N spanned by solutions of the governing partial differential 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 nonlinearity exercises symmetry factorization procedures necessary for treatment of high-order Galerkin summations in the (say) residual dual-norm calculation; and the Burgers equation 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.

Certified Reduced Basis Methods for Parametrized Partial Differential Equations

“…While the former is directly applicable in the multi-dimensional parameter domain, the latter is most often applied only in the one-dimensional Reduced basis methods can be effectively applied also to nonlinear problems [51, 17,74], although this typically introduces both numerical and theoretical complications, and many questions remain open. For classical problems with a quadratic nonlinearity, there has been substantial progress, e.g., Navier-Stokes/Boussinesq and Burgers' equations in fluid mechanics [123,156,155,31,128,35,148] and nonlinear elasticity in solid mechanics.…”

“…However, the development of effective sampling strategies, in particular in the case of many parameters [31,138,156,103], can also be aided by the error estimators. These can play an important role in the development of efficient and effective sampling procedures by utilizing inexpensive error bounds to explore much larger subsets of the parameter domain in search of the most representative snapshots, and to determine when the basis is sufficiently rich.…”

“…Hence, to capture the causality associated with the evolution, the sampling method combines the proper orthogonal decomposition, see Section 3.2.1, for the time-trajectories, with the greedy procedure in the parameter space [52,156,144] to enable the efficient treatment of the higher dimensions and extensive ranges of parameter variation.…”

Solving Parameter-Dependent Lyapunov Equations Using the Reduced Basis Method with Application to Parametric Model Order Reduction

“…This method was initially proposed for coercive elliptic partial differ-ential equations and then extended to non-coercive equations [32], Burgers equations [31], and Navier-Stokes equations [8]. Applying this method to PALEs is most probably the first time performed in the present paper, while an extension to parametric Riccati equation can be found in a very recent paper [12].…”

A reduced basis element method for the steady Stokes problem: Application to hierarchical flow systems

“…We follow the method presented in Veroy et al (2003), where the output bound gap developed in Section 4 is used to reorder the basis functions, such that the error in the output of interest, s(u), is minimized. We also recall that S N is the set of preselected parameter values.…”