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

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 a new procedure and attendant theoretical foundations for adaptive partition of the parameter domain into parameter subdomains (“h”-refinement); subsequently, we construct individual standard reduced basis approximation spaces for each subdomain (“p”-refinement). Greedy parameter sampling procedures and a posteriori error estimation are the main ingredients of the new algorithm. We present illustrative numerical results for a convectiondiffusion 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.