Harald van Brummelen

Eivind Fonn, Harald van Brummelen, Trond Kvamsdal et al.

Reduced-basis methods (RB methods or RBMs) form one of the most promising techniques to deliver numerical solutions of parametrized PDEs in real-time performance with reasonable accuracy. For incompressible flow problems, RBMs based on LBB stable velocity-pressure spaces do not generally inherit the stability of the underlying high-fidelity model and, instead, additional stabilization techniques must be introduced. One way of bypassing the loss of LBB stability in the RBM is to inflate the velocity space with supremizer modes. This however deteriorates the performance of the RBM in the performance-critical online stage, as additional DOFs must be introduced to retain stability, while these DOFs do not effectively contribute to accuracy of the RB approximation. In this work we consider a velocity-only RB approximation, exploiting a solenoidal velocity basis. The solenoidal reduced basis emerges directly from the high-fidelity velocity solutions in the offline stage. By means of Piola transforms, the solenoidality of the velocity space is retained under geometric transformations, making the proposed RB method suitable also for the investigation of geometric parameters. To ensure exact solenoidality of the high-fidelity velocity solutions that constitute the RB, we consider approximations based on divergence-conforming compatible B-splines. We show that the velocity-only RB method leads to a significant improvement in computational efficiency in the online stage, and that the pressure solution can be recovered a posteriori at negligible extra cost. We illustrate the solenoidal RB approach by modeling steady two-dimensional Navier-Stokes flow around a NACA0015 airfoil at various angles of attack.

Frits de Prenter, Clemens Verhoosel, Harald van Brummelen

Immersed finite element methods generally suffer from conditioning problems when cut elements intersect the physical domain only on a small fraction of their volume. De Prenter et al. [Computer Methods in Applied Mechanics and Engineering, 316 (2017) pp. 297-327] present an analysis for symmetric positive definite (SPD) immersed problems, and for this class of problems an algebraic preconditioner is developed. In this contribution the conditioning analysis is extended to immersed finite element methods for systems that are not SPD and the preconditioning technique is generalized to a connectivity-based preconditioner inspired by Additive-Schwarz preconditioning. This Connectivity-based Additive-Schwarz (CbAS) preconditioner is applicable to problems that are not SPD and to mixed problems, such as the Stokes and Navier-Stokes equations. A detailed numerical investigation of the effectivity of the CbAS preconditioner to a range of flow problems is presented.

Xunxun Wu, Kristoffer van der Zee, Gorkem Simsek et al.

While many methods exist to discretize nonlinear time-dependent partial differential equations (PDEs), the rigorous estimation and adaptive control of their discretization errors remains challenging. In this paper, we present a methodology for duality-based a posteriori error estimation for nonlinear parabolic PDEs, where the full discretization of the PDE relies on the use of an implicit-explicit (IMEX) time-stepping scheme and the finite element method in space. The main result in our work is a decomposition of the error estimate that allows to separate the effects of spatial and temporal discretization error, and which can be used to drive adaptive mesh refinement and adaptive time-step selection. The decomposition hinges on a specially-tailored IMEX discretization of the dual problem. The performance of the error estimates and the proposed adaptive algorithm is demonstrated on two canonical applications: the elementary heat equation and the nonlinear Allen-Cahn phase-field model.