We present a new approach for the construction of lower bounds for the inf-sup stability constants required in a posteriori error analysis of reduced basis approximations to affinely parametrized partial differential equations. We combine the "linearized" inf-sup statement of the naturalnorm approach with the approximation procedure of the Successive Constraint Method (SCM): the former (natural-norm) provides an economical parameter expansion and local concavity in parameter -a small(er) optimization problem which enjoys intrinsic lower bound properties; the latter (SCM) provides a systematic optimization framework -a Linear Program (LP) relaxation which readily incorporates continuity and stability constraints. The natural-norm SCM requires a parameter domain decomposition: we propose a greedy algorithm for selection of the SCM control points as well as adaptive construction of the optimal subdomains. The efficacy of the natural-norm SCM is illustrated through numerical results for two types of non-coercive problems: the Helmholtz equation (for acoustics, elasticity, and electromagnetics), and the convection-diffusion equation.
In this paper we present reduced basis approximations and associated rigorous a posteriori error bounds for the parametrized unsteady Boussinesq equations. The essential ingredients are Galerkin projection onto a low-dimensional space associated with a smooth parametric manifold -to provide dimension reduction; an efficient POD-Greedy sampling method for identification of optimal and numerically stable approximationsto yield rapid convergence; accurate (Online) calculation of the solution-dependent stability factor by the Successive Constraint Method -to quantify the growth of perturbations/residuals in time; rigorous a posteriori bounds for the errors in the reduced basis approximation and associated outputs -to provide certainty in our predictions; and an Offline-Online computational decomposition strategy for our reduced basis approximation and associated error bound -to minimize marginal cost and hence achieve high performance in the real-time and many-query contexts. The method is applied to a transient natural convection problem in a two-dimensional "complex" enclosure -a square with a small rectangle cut-out -parametrized by Grashof number and orientation with respect to gravity. Numerical results indicate that the reduced basis approximation converges rapidly and that furthermore the (inexpensive) rigorous a posteriori error bounds remain practicable for parameter domains and final times of physical interest.
Abstract. This paper is concerned with the analysis and implementation of spectral Galerkin methods for a class of Fokker-Planck equations that arises from the kinetic theory of dilute polymers. A relevant feature of the class of equations under consideration from the viewpoint of mathematical analysis and numerical approximation is the presence of an unbounded drift coefficient, involving a smooth convex potential U that is equal to +∞ along the boundary ∂D of the computational domain D. Using a symmetrization of the differential operator based on the Maxwellian M corresponding to U , which vanishes along ∂D, we remove the unbounded drift coefficient at the expense of introducing a degeneracy, through M , in the principal part of the operator. The general class of admissible potentials considered includes the FENE (finitely extendible nonlinear elastic) model. We show the existence of weak solutions to the initial-boundary-value problem, and develop a fully-discrete spectral Galerkin method for such degenerate Fokker-Planck equations that exhibits optimal-order convergence in the Maxwellian-weighted H 1 norm on D. In the case of the FENE model, we also discuss variants of these analytical results when the Fokker-Planck equation is subjected to an alternative class of transformations proposed by Chauvière and Lozinski; these map the original Fokker-Planck operator with an unbounded drift coefficient into Fokker-Planck operators with unbounded drift and reaction coefficients, that have improved coercivity properties in comparison with the original operator. The analytical results are illustrated by numerical experiments for the FENE model in two space dimensions.
Mathematics Subject Classification.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.