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Archive : Numerical Analysis and PDE

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Finite Element CircusFinite Element CircusZoom<a href="https://sites.google.com/view/fecircus">Event Information </a>4/9/2021 4:00:00 PM4/10/2021 5:00:00 PMFalse
Jingwei Hu, Purdue University. Jingwei Hu, Purdue University. Zoom:<br> <br> Title: An efficient dynamical low-rank algorithm for the Boltzmann-BGK equation close to the compressible viscous flow regime <br> <br> Abstract: It has recently been demonstrated that dynamical low-rank algorithms can provide robust and efficient approximation to a range of kinetic equations. This is true especially if the solution is close to some asymptotic limit where it is known that the solution is low-rank. A particularly interesting case is the fluid dynamic limit that is commonly obtained in the limit of small Knudsen number. However, in this case the Maxwellian which describes the corresponding equilibrium distribution is not necessarily low-rank; because of this, the methods known in the literature are only applicable to the weakly compressible case. In this paper, we propose an efficient dynamical low-rank integrator that can capture the fluid limit -- the Navier-Stokes equations -- of the Boltzmann-BGK model even in the compressible regime. This is accomplished by writing the solution as f=Mg, where M is the Maxwellian and the low-rank approximation is only applied to g. To efficiently implement this decomposition within a low-rank framework requires that certain coefficients are evaluated using convolutions, for which fast algorithms are known. Using the proposed decomposition also has the advantage that the rank required to obtain accurate results is significantly reduced compared to the previous state of the art. We demonstrate this by performing a number of numerical experiments and also show that our method is able to capture sharp gradients/shock waves. This is joint work with Lukas Einkemmer (Innsbruck) and Lexing Ying (Stanford).4/2/2021 3:00:00 PM4/2/2021 4:00:00 PMFalse
Daniel Appelö, Michigan State UniversityDaniel Appelö, Michigan State UniversityZoom<br> Title: WaveHoltz: Parallel and Scalable Solution of the Helmholtz Equation via Wave Equation Iteration <br> Abstract: We introduce a novel idea, the WaveHoltz iteration, for solving the Helmholtz equation. Our method make use of time domain methods for wave equations to design frequency domain Helmholtz solvers. We show that the WaveHoltz iteration we propose results in a symmetric and positive definite linear system even though we are solving the Helmholtz equation. As our method utilizes time-domain solvers we can exploit features such as local timesteping that are not present in the frequency domain. As this seminar is just before lunch we will also show a unique “free lunch” property that allows us to solve for multiple wave numbers at at the cost of a single solve. We will present numerical examples, using various discretization techniques, that show that our method can be used to solve problems with rather high wave numbers. This is joint work with, Fortino Garcia, University of Colorado, Boulder, USA and Olof Runborg, Royal Institute of Technology, Stockholm, Sweden.2/19/2021 4:00:00 PM2/19/2021 5:00:00 PMFalse
Jake Jacavage, Lafayette CollegeJake Jacavage, Lafayette CollegeZoomTitle: A Least Squares Discretization for Parametric Mixed Variational Formulations <br> Abstract: Mixed variational formulations for partial differential equations arise naturally when modeling physical systems. When approximating the physical quantities from these systems, it is beneficial to obtain estimates that are robust with respect to the parameters associated with the system. In this talk, we will summarize some general ideas regarding an approach to approximating solutions of mixed variational problems using a saddle point reformulation. We will also discuss the concepts of optimal and almost optimal test norm as well as provide estimates for the continuity and stability constants. Applications of the method will include second order elliptic interface problems as well as reaction-diffusion problems. 12/11/2020 4:00:00 PM12/11/2020 5:00:00 PMFalse

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