POD model order reduction with space-adapted snapshots for incompressible flows

We consider model order reduction based on proper orthogonal decomposition (POD) for unsteady incompressible Navier-Stokes problems, assuming that the snapshots are given by spatially adapted finite element solutions. We propose two approaches of deriving stable POD-Galerkin reduced-order models for this context. In the first approach, the pressure term and the continuity equation are eliminated by imposing a weak incompressibility constraint with respect to a pressure reference space. In the second approach, we derive an inf-sup stable velocity-pressure reduced-order model by enriching the velocity reduced space with supremizers computed on a velocity reference space. For problems with inhomogeneous Dirichlet conditions, we show how suitable lifting functions can be obtained from standard adaptive finite element computations. We provide a numerical comparison of the considered methods for a regularized lid-driven cavity problem

New POD Error Expressions, Error Bounds, and Asymptotic Results for Reduced Order Model of Parabolic PDEs

The derivations of existing error bounds for reduced order models of time varying partialdi erential equations (PDEs) constructed using proper orthogonal decomposition (POD) haverelied on bounding the error between the POD data and various POD projections of that data.Furthermore, the asymptotic behavior of the model reduction error bounds depends on theasymptotic behavior of the POD data approximation error bounds. We consider time varyingdata taking values in two di erent Hilbert spacesHandV, withVH, and prove exactexpressions for the POD data approximation errors considering four di erent POD projectionsand the two di erent Hilbert space error norms. Furthermore, the exact error expressions canbe computed using only the POD eigenvalues and modes, and we prove the errors converge tozero as the number of POD modes increases. We consider the POD error estimation approachesof Kunisch and Volkwein (SIAM J. Numer. Anal., 40, pp. 492-515, 2002) and Chapelle, Gariah,and Sainte-Marie (ESAIM Math. Model. Numer. Anal., 46, pp. 731-757, 2012) and apply ourresults to derive new POD model reduction error bounds and convergence results for the twodimensional Navier-Stokes equations. We prove the new error bounds tend to zero as the numberof POD modes increases for POD spaceX=Hin both approaches; the asymptotic behaviorof existing error bounds was unknown for this case. Also, forX=H, we prove one new errorbound tends to zero without requiring time derivative data in the POD data set

Convergent finite element discretizations of the nonstationary incompressible magnetohydrodynamics system

The incompressible MHD equations couple Navier-Stokes equations with Maxwell's equations to describe the flow of a viscous, incompressible, and electrically conducting fluid in a Lipschitz domain $\Omega \subset \mathbb{R}^3$. We verify convergence of iterates of different coupling and decoupling fully discrete schemes towards weak solutions for vanishing discretization parameters. Optimal first order of convergence is shown in the presence of strong solutions for a splitting scheme which decouples the computation of velocity field, pressure, and magnetic fields at every iteration step

Spectral methods for CFD

One of the objectives of these notes is to provide a basic introduction to spectral methods with a particular emphasis on applications to computational fluid dynamics. Another objective is to summarize some of the most important developments in spectral methods in the last two years. The fundamentals of spectral methods for simple problems will be covered in depth, and the essential elements of several fluid dynamical applications will be sketched