11 March – Omar Duran: Explicit and implicit hybrid high-order methods for the wave equation in time regime

Omar Duran: Thursday 11 March at 15:00 There are two main approaches to derive a fully discrete method for solving the second-order acoustic wave equation in the time regime. One is to discretize directly the second-order time derivative and the Laplacian operator in space. The other approach is to transform the second-order equation into a first-order hyperbolic system. Firstly, we consider the time second-order form. We devise, analyze the energy-conservation properties, and evaluate numerically a hybrid high-order (HHO) scheme for the space discretization combined with a Newmark-like time-marching scheme. The HHO method uses as discrete unknowns cell- and face-based polynomials of some order 0 ≤ k, yielding for steady problems optimal convergence of order (k + 1) in the energy norm [1]. Secondly, inspired by ideas presented in [2] for hybridizable discontinuous Galerkin (HDG) method and the link between HDG and HHO methods in the steady case [3], first-order explicit or implicit time-marching schemes combined with the HHO method for space discretization are considered. We discuss the selection of the stabilization term and energy conservation and present numerical examples. Extension to the unfitted meshes is contemplated for the acoustic wave equation. We observe that the unfitted approach combined with local cell agglomeration leads to a comparable CFL condition as when using fitted meshes [4]. [1] D.A. Di Pietro, A. Ern, and S. Lemaire. An arbitrary-order and compact-stencil discretization of diffusion on general meshes based on local reconstruction operators. Computational Methods in Applied Mathematics. 14 (2014) 461-472. [2] M. Stanglmeier, N.C. Nguyen, J. Peraire, and B. Cockburn. An explicit hybridizable discontinuous Galerkin method for the acoustic wave equation. Computer Methods in Applied Mechanics and Engineering, 300:748–769, March 2016. [3] B. Cockburn, D. A. Di Pietro, and A. Ern. Bridging the hybrid high-order and hybridizable discontinuous Galerkin methods. ESAIM: Mathematical Modelling and…

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25 February – Buyang Li : A bounded numerical solution with a small mesh size implies existence of a smooth solution to the time-dependent Navier–Stokes equations

Buyang Li: Thursday 25 February at 14:00 We prove that for a given smooth initial value, if one finite element solution of the three-dimensional time-dependent Navier–Stokes equations is bounded by $M$ when some sufficiently small step size $\tau < \tau _M$ and mesh size $h < h_M$ are used, then the true solution of the Navier–Stokes equations with this given initial value must be smooth and unique, and is successfully approximated by the numerical solution.

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18 February – Roland Maier : Multiscale scattering in nonlinear Kerr-type media

18 February – Roland Maier Wave propagation in heterogeneous and nonlinear media has arisen growing interest in the last years since corresponding materials can lead to unusual and interesting effects and therefore come with a wide range of applications. An important example of such materials is Kerr-type media, where the intensity of a wave directly influences the refractive index. In the time-harmonic regime, this effect can be modeled with a nonlinear Helmholtz equation. If underlying material coefficients are highly oscillatory on a microscopic scale, the numerical approximation of corresponding solutions can be a delicate task. In this talk, a multiscale technique is presented that allows one to deal with microscopic coefficients in a nonlinear Helmholtz equation without the need for global fine-scale computations. The method is based on an iterative and adaptive construction of appropriate multiscale spaces based on the multiscale method known as Localized Orthogonal Decomposition, which works under minimal structural assumptions. This talk is based on joint work with Barbara Verfürth (KIT, Karlsruhe)  

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