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

Abstract

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.

Keywords:

• parabolic partial differential equations
• certified reduced basis
• a posteriori error estimation
• POD/Greedy
• hp reduced basis
• convection–diffusion
• Boussinesq natural convection

Acknowledgements

This work was supported by the Norwegian University of Science and Technology, AFOSR Grant No. FA9550-07-1-0425, and OSD/AFOSR Grant No. FA9550-09-1-0613.

Notes

¹ In the linear case , and it thus follows from Equation (9)(9) and the definition of (we recall that is coercive) that Equation (10)(10) simplifies to .

² We refer to [12,15] for details on the Construction–Evaluation procedure for the computation of lower bounds for the stability constants – a Successive Constraint Method (SCM).

³ We note that .

⁴ We note that after completion of the hp-POD/Greedy, we can apply the SCM algorithm independently for each parameter subdomain; we thus expect a reduction in the SCM (Online) evaluation cost because the size of the parameter domain is effectively reduced.

⁵ We suppose here for simplicity that ; hence for all .

⁶ In fact, we should interpret M here as the number of subdomains generated by Algorithm 4.3 so far; the , , are not necessarily the final M subdomains. With this interpretation, we thus do not presume termination of the algorithm.

⁷ Equation (16)(16) is satisfied with . We note that is continuous in its second argument because by the Cauchy–Schwarz inequality .

⁸ To ensure a good spread over of the rather few (25 or 64 for our two examples) initial train points, we use for a deterministic initial regular grid. (For the train sample enrichment, we use random points.) As some train points belong to a regular grid, the procedure may produce ‘aligned’ subdomain boundaries, as seen in Figure 3.

⁹ We note that .