A generalized finite element method for the strongly damped wave equation with rapidly varying data

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, 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, independent of the derivatives of the coefficients. We present numerical examples that confirm the theoretical findings.

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Mixed finite element method for a strongly damped wave equation

Numerical Methods for Partial Differential Equations ◽

10.1002/1098-2426(200103)17:2<105::aid-num2>3.0.co;2-f ◽

2001 ◽

Vol 17 (2) ◽

pp. 105-119 ◽

Cited By ~ 10

Author(s):

Amiya K. Pani ◽

Jin Yun Yuan

Keyword(s):

Finite Element Method ◽

Finite Element ◽

Wave Equation ◽

Mixed Finite Element ◽

Mixed Finite Element Method ◽

Damped Wave Equation ◽

Strongly Damped Wave Equation ◽

Strongly Damped ◽

Element Method

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Error estimates of finite element method for semilinear stochastic strongly damped wave equation

IMA Journal of Numerical Analysis ◽

10.1093/imanum/dry030 ◽

2018 ◽

Vol 39 (3) ◽

pp. 1594-1626 ◽

Cited By ~ 1

Author(s):

Ruisheng Qi ◽

Xiaojie Wang

Keyword(s):

Finite Element Method ◽

Finite Element ◽

Wave Equation ◽

Error Estimates ◽

Mild Solution ◽

Space Time ◽

Damped Wave Equation ◽

Strongly Damped Wave Equation ◽

Strongly Damped ◽

Element Method

AbstractIn this paper we consider a semilinear stochastic strongly damped wave equation driven by additive Gaussian noise. Following a semigroup framework we establish existence, uniqueness and space-time regularity of a mild solution to such equation. Unlike the usual stochastic wave equation without damping the underlying problem with space-time white noise (Q = I) allows for a mild solution with a positive order of regularity in multiple spatial dimensions. Further, we analyze a spatio-temporal discretization of the problem, performed by a standard finite element method (FEM) in space and a well-known linear implicit Euler scheme in time. The analysis of the approximation error forces us to significantly enrich existing error estimates of semidiscrete and fully discrete FEMs for the corresponding linear deterministic equation. The main results show optimal convergence rates in the sense that the orders of convergence in space and in time coincide with the orders of the spatial and temporal regularity of the mild solution, respectively. Numerical examples are finally included to confirm our theoretical findings.

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Asymptotic Expansions and Extrapolations of H1-Galerkin Mixed Finite Element Method for Strongly Damped Wave Equation

Advances in Applied Mathematics and Mechanics ◽

10.4208/aamm.2013.m90 ◽

2015 ◽

Vol 7 (5) ◽

pp. 610-624 ◽

Cited By ~ 3

Author(s):

Dongyang Shi ◽

Qili Tang ◽

Xin Liao

Keyword(s):

Finite Element Method ◽

Finite Element ◽

Wave Equation ◽

Convergence Rates ◽

Mixed Finite Element ◽

Mixed Finite Element Method ◽

Damped Wave Equation ◽

Strongly Damped Wave Equation ◽

Strongly Damped ◽

Element Method

AbstractIn this paper, a high-accuracy H1-Galerkin mixed finite element method (MFEM) for strongly damped wave equation is studied by linear triangular finite element. By constructing a suitable extrapolation scheme, the convergence rates can be improved from 𝒪(h) to 𝒪(h3) both for the original variable u in H1(Ω) norm and for the actual stress variable p = ∇ut in H(div;Ω) norm, respectively. Finally, numerical results are presented to confirm the validity of the theoretical analysis and excellent performance of the proposed method.

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