Abstract
We propose a generalized finite element method for the strongly damped wave equation with highly varying coefficients. The proposed method is based on the localized orthogonal decomposition introduced in Målqvist and Peterseim [Math. Comp. 83 (2014) 2583–2603], and is designed to handle independent variations in both the damping and the wave propagation speed respectively. The method does so by automatically correcting for the damping in the transient phase and for the propagation speed in the steady state phase. Convergence of optimal order is proven in L2(H1)-norm, independent of the derivatives of the coefficients. We present numerical examples that confirm the theoretical findings.
Highlights
This paper is devoted to the study of numerical solutions to the strongly damped wave equation with highly varying coefficients
A and B represent the system’s damping and wave propagation respectively, f denotes the source term, and the solution u is a displacement function. This equation commonly appears in the modelling of viscoelastic materials, where the strong damping −∇ · A∇uarises due to the stress being represented as the sum of an elastic part and a viscous part [6, 13]
We propose a novel multiscale method based on the localized orthogonal decomposition (LOD) method
Summary
This paper is devoted to the study of numerical solutions to the strongly damped wave equation with highly varying coefficients. Multiscale, localized orthogonal decomposition, finite element method, reduced basis method. Multiscale methods, as the localized orthogonal decomposition, are usually designed to handle problems with a single multiscale coefficient In this sense, the strongly damped wave equation is different, as an extra coefficient appears due to the strong damping. In this paper we present a generalized finite element method (GFEM), with a backward Euler time stepping for solving the strongly damped wave equation. The method uses both the damping and diffusion coefficients to construct a generalized finite element space, similar to those in e.g.
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callMore From: ESAIM: Mathematical Modelling and Numerical Analysis
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
