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

• Main Authors: Eftang, Jens L. (Author), Patera, Anthony T. (Contributor), Ronquist, Einar M. (Author)
• Other Authors: Massachusetts Institute of Technology. Department of Mechanical Engineering (Contributor)
• Format: Article
• Language: English
• Published: Society for Industrial and Applied Mathematics, 2010-09-03T20:43:14Z.
• Subjects: Article
• Online Access: Get fulltext

 We present a new "hp" parameter multi-domain certified reduced basis 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.