Reduced-order modeling of time-dependent PDEs with multiple parameters in the boundary data.

*(English)* Zbl 1121.65354
Summary: The computational approximation of solutions to nonlinear partial differential equations (PDEs) such as the Navier-Stokes equations is often a formidable task. For this reason, there is significant interest in the development of very low-dimensional models that can be used to determine reasonably accurate approximations in simulation and control problems for PDEs. Many concrete tests have been reported on in the literature that show that several classes of reduced-order models (ROMs) are effective, i.e., an accurate approximate solution can be obtained using very low-dimensional ROMs. Most of these tests involved problems for which the solution depends on only a single parameter appearing in the boundary data. The extension of the techniques used in those tests to the case of multiple parameters is not a straightforward matter. Here, we present, test, and compare two methods for treating, within the context of a class of ROMs, inhomogeneous Dirichlet-type boundary conditions that contain multiple parameters. In this study, we focus on the proper orthogonal decomposition (POD) approach to ROM; however, the issues and results discussed here in the POD context apply equally well to other ROM approaches.

##### MSC:

65M60 | Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (IVP of PDE) |