scholarly journals On a singularly perturbed wave equation with dynamic boundary conditions

2004 ◽  
Vol 134 (2) ◽  
pp. 389-413 ◽  
Author(s):  
Luminiţa Popescu ◽  
Aníbal Rodriguez-Bernal
Keyword(s):  
Boundary Conditions ◽  
Wave Equations ◽  
Limit Problem ◽  
Linear Wave ◽  
Dynamic Boundary Conditions ◽  
Boundary Damping ◽  
Perturbation Problem ◽  
Dynamic Boundary ◽  
Linear Wave Equations ◽  
Parabolic Limit

In this paper we analyse a singular perturbation problem for linear wave equations with interior and boundary damping. We show how the solutions converge to the formal parabolic limit problem with dynamic boundary conditions. Conditions are given for uniform convergence in the energy space.

scholarly journals Finite element error analysis of wave equations with dynamic boundary conditions: L2 estimates

2020 ◽  
Author(s):  
David Hipp ◽  
Balázs Kovács
Keyword(s):  
Boundary Conditions ◽  
Error Estimates ◽  
Wave Equations ◽  
Linear Wave ◽  
Dynamic Boundary Conditions ◽  
Surface Finite Elements ◽  
Dynamic Boundary ◽  
Bulk Surface ◽  
Spatial Convergence ◽  
Element Error

Abstract $L^2$ norm error estimates of semi- and full discretizations of wave equations with dynamic boundary conditions, using bulk–surface finite elements and Runge–Kutta methods, are studied. The analysis rests on an abstract formulation and error estimates, via energy techniques, within this abstract setting. Four prototypical linear wave equations with dynamic boundary conditions are analysed, which fit into the abstract framework. For problems with velocity terms or with acoustic boundary conditions we prove surprising results: for such problems the spatial convergence order is shown to be less than 2. These can also be observed in the presented numerical experiments.

scholarly journals General decay and blow-up for coupled Kirchhoff wave equations with dynamic boundary conditions

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Mengxian Lv ◽  
Jianghao Hao
Keyword(s):  
Boundary Conditions ◽  
Wave Equations ◽  
Blow Up ◽  
Initial Energy ◽  
Decay Rates ◽  
General Decay ◽  
Dynamic Boundary Conditions ◽  
Positive Initial Energy ◽  
Relaxation Functions ◽  
Dynamic Boundary

<p style='text-indent:20px;'>In this paper we consider a system of viscoelastic wave equations of Kirchhoff type with dynamic boundary conditions. Supposing the relaxation functions <inline-formula><tex-math id="M1">\begin{document}$ g_i $\end{document}</tex-math></inline-formula> <inline-formula><tex-math id="M2">\begin{document}$ (i = 1, 2, \cdots, l) $\end{document}</tex-math></inline-formula> satisfy <inline-formula><tex-math id="M3">\begin{document}$ g_i(t)\leq-\xi_i(t)G(g_i(t)) $\end{document}</tex-math></inline-formula> where <inline-formula><tex-math id="M4">\begin{document}$ G $\end{document}</tex-math></inline-formula> is an increasing and convex function near the origin and <inline-formula><tex-math id="M5">\begin{document}$ \xi_i $\end{document}</tex-math></inline-formula> are nonincreasing, we establish some optimal and general decay rates of the energy using the multiplier method and some properties of convex functions. Moreover, we obtain the finite time blow-up result of solution with nonpositive or arbitrary positive initial energy. The results in this paper are obtained without imposing any growth condition on weak damping term at the origin. Our results improve and generalize several earlier related results in the literature.</p>

scholarly journals Decoupling techniques for wave equations with dynamic boundary conditions

2005 ◽  
Vol 12 (4) ◽  
pp. 761-772 ◽  
Author(s):  
V. Casarino ◽  
◽  
K.-J. Engel ◽  
G. Nickel ◽  
S. Piazzera ◽  
...  
Keyword(s):  
Boundary Conditions ◽  
Wave Equations ◽  
Dynamic Boundary Conditions ◽  
Dynamic Boundary
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