An "$hp$" Certified Reduced Basis Method for Parametrized Elliptic Partial Differential Equations

Authors:
Jens L. Eftang;Anthony T. Patera;Einar M. Rønquist
Affiliations:
eftang@math.ntnu.no and ronquist@math.ntnu.no;patera@mit.edu;-
Venue:
SIAM Journal on Scientific Computing
Year:
2010

Citing 2
Cited 5

On the theory and error estimation of the reduced basis method for multi-parameter problems

Nonlinear Analysis: Theory, Methods & Applications
A Certified Reduced Basis Method for the Fokker-Planck Equation of Dilute Polymeric Fluids: FENE Dumbbells in Extensional Flow

SIAM Journal on Scientific Computing

The reduced basis method for the electric field integral equation

Journal of Computational Physics
Reduced Basis Method and Error Estimation for Parametrized Optimal Control Problems with Control Constraints

Journal of Scientific Computing
A Two-Step Certified Reduced Basis Method

Journal of Scientific Computing
Reduced Basis Approximation for Nonlinear Parametrized Evolution Equations based on Empirical Operator Interpolation

SIAM Journal on Scientific Computing
Two-Step Greedy Algorithm for Reduced Order Quadratures

Journal of Scientific Computing

Quantified Score

Hi-index	0.01

Visualization

Abstract

We present a new “$hp$” parameter multidomain certified reduced basis (RB) method for rapid and reliable online evaluation of functional outputs associated with parametrized elliptic partial differential equations. We propose, and provide theoretical justification for, a new procedure for adaptive partition (“$h$”-refinement) of the parameter domain into smaller parameter subdomains: we pursue a hierarchical splitting of the parameter (sub)domains based on proximity to judiciously chosen parameter anchor points within each subdomain. Subsequently, we construct individual standard RB approximation spaces (“$p$”-refinement) over each subdomain. Greedy parameter sampling procedures and a posteriori error estimation play important roles in both the “$h$”-type and “$p$”-type stages of the new algorithm. We present illustrative numerical results for a convection-diffusion problem: the new “$hp$”-approach is considerably faster (respectively, more costly) than the standard “$p$”-type reduced basis method in the online (respectively, offline) stage.