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# Numerical Analysis and PDE

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All talks will be online, on Fridays from 11 a.m. to noon, followed by a 30-minutes interaction with the speaker on zoom. The Zoom room will be available from 10:30 a.m., and requires a passcode to access. The passcode will be sent in the weekly seminar announcement. If you are interested, but do not have access to the passcode, please contact Dr. Qiu, jingqiu@udel.edu​​

Want to suggest a speaker? Send email to the organizer, Jingmei Qiu.

Information about the Courtyard Newark - University of Delaware Hotel.​

 Yongtao Zhang, Notre Dame University 146 Yongtao Zhang, Notre Dame University Zoom

Title: Fast sparse grid simulations of high order WENO schemes
Speaker: Yongtao Zhang, University of Notre Dame
Abstract: Mathematical models arising from applications are often PDE problems defined on multidimensional spatial domains. However, when the spatial dimension increases, the number of spatial grid points increases significantly. It leads to large amount of operations and computational costs in the numerical simulations, especially for high order accuracy schemes. In the literature, sparse-grid technique has been developed as a very efficient approximation tool for high dimensional problems. In this talk, I will present our recent results on designing sparse grid weighted essentially non-oscillatory (WENO) schemes for solving hyperbolic PDEs. Our goal is to apply sparse grid techniques in high order WENO schemes to achieve more efficient computations than that in their regular performance in solving multidimensional PDEs. A challenge is how to design the schemes on sparse grids such that comparable high order accuracy of the WENO schemes in smooth regions of the solutions can still be achieved as that for computations on regular single grids. Additional challenge is that nonlinear stability of WENO simulations needs to be preserved. We apply sparse-grid combination approach to overcome these difficulties. WENO interpolations are applied for the prolongation part in sparse-grid combination technique. The approach is applied to both classical WENO scheme and the fifth order multiresolution WENO scheme which is much simpler in computing linear weights. Numerical examples defined on domains with up to four spatial dimensions are presented to show that significant computational times are saved, while both accuracy and stability of the original schemes are maintained for numerical simulations on sparse grids. The methods are also applied to some multidimensional Vlasov based PDEs to further demonstrate large savings of computational costs by comparing with simulations on regular single grids. This is joint work with graduate students Ernie Tsybulnik and Xiaozhi Zhu at Notre Dame.

10/22/2021 3:00:00 PM 10/22/2021 4:00:00 PM Dr. Xue Hong, University of Delaware 149 Dr. Xue Hong, University of Delaware 10/29/2021 3:00:00 PM 10/29/2021 4:00:00 PM Mark Gockenbach, University of Delaware 147 Mark Gockenbach, University of Delaware Zoom 11/12/2021 4:00:00 PM 11/12/2021 5:00:00 PM Yangwen Zhang, Carnegie Mellon University 145 Yangwen Zhang, Carnegie Mellon University Zoom

Title: Sharp $L^\infty$ estimates of HDG methods for Poisson equation II: 3D
Abstract: In [SIAM J. Numer. Anal., 59 (2), 720-745], we proved quasi-optimal $L^\infty$ estimates (up to logarithmic factors) for the solution of Poisson's equation by a hybridizable discontinuous Galerkin (HDG) method. However, the estimates \emph{only} work in 2D. In this paper, we use the approach in [Numer. Math., 131 (2015), pp. 771–822] and obtain \emph{sharp} (without logarithmic factors) $L^\infty$ estimates for the HDG method in both 2D and 3D. Numerical experiments are presented to confirm our theoretical result.

11/19/2021 4:00:00 PM 11/19/2021 5:00:00 PM

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