David J. Knezevic

An hp certified reduced basis method for parametrized parabolic partial differential equations

In this article, we introduce an hp certified reduced basis (RB) method for parabolic partial differential equations. We invoke a Proper Orthogonal Decomposition (POD) (in time)/Greedy (in parameter) sampling procedure first in the initial partition of the parameter domain (h-refinement) and subsequently in the construction of RB approximation spaces restricted to each parameter subdomain (p-refinement). We show that proper balance between additional POD modes and additional parameter values in the initial subdivision process guarantees convergence of the approach. We present numerical results for two model problems: linear convection–diffusion and quadratically non-linear Boussinesq natural convection. The new procedure is significantly faster (more costly) in the RB Online (Offline) stage.

A Certified Reduced Basis Method for the Fokker-Planck Equation of Dilute Polymeric Fluids: FENE Dumbbells in Extensional Flow

In this paper we present a reduced basis method for the parametrized Fokker-Planck equation associated with evolution of finitely extensible nonlinear elastic (FENE) dumbbells in a Newtonian solvent for a (prescribed) extensional macroscale flow. We apply a proper orthogonal decomposition (POD)-greedy sampling procedure for the stable identification of optimal reduced basis spaces, and we develop a rigorous finite-time a posteriori bound for the error in the reduced basis prediction of the two outputs of interest—the optical anisotropy and the first normal stress difference. We present numerical results for stress-conformation hysteresis as a function of Weissenberg number and final time that demonstrate the rapid convergence of the reduced basis approximation and the effectiveness of the a posteriori error bounds.

A Two-Step Certified Reduced Basis Method

In this paper we introduce a two-step Certified Reduced Basis (RB) method. In the first step we construct from an expensive finite element “truth” discretization of dimension

Adaptive Port Reduction in Static Condensation

{\mathcal{N}}

Reduced basis approximation and a posteriori error estimates for a multiscale liquid crystal model

an intermediate RB model of dimension

REDUCED BASIS APPROXIMATION AND A POSTERIORI ERROR ESTIMATION FOR THE PARAMETRIZED UNSTEADY BOUSSINESQ EQUATIONS

N\ll {\mathcal{N}}

A Static condensation Reduced Basis Element method : approximation and a posteriori error estimation

. In the second step we construct from this intermediate RB model a derived RB (DRB) model of dimension M≤N. The construction of the DRB model is effected at cost

A natural-norm Successive Constraint Method for inf-sup lower bounds

{\mathcal{O}}(N)

Methods and apparatus for constructing and analyzing component-based models of engineering systems

and in particular at cost independent of

A high-performance parallel implementation of the certified reduced basis method

{\mathcal{N}}