In this article, we present an error estimation in the norm referring to three wave models with variable coefficients, supplemented with initial and boundary conditions. The first two models are nonlinear wave equations with Dirichlet, Acoustics, and nonlinear dissipative impenetrability boundary conditions, while the third model is a linear wave equation with Dirichlet, Acoustics, and linear dissipative impenetrability boundary conditions. In the field of numerical analysis, we establish two key theorems for estimating errors to the semi-discrete and totally discrete problems associated with each model. Such theorems provide theoretical results on the convergence rate in both space and time. For conducting numerical simulations, we employ linear, quadratic, and cubic polynomial basis functions for the finite element spaces in the Galerkin method, in conjunction with the Crank-Nicolson method for time discretization. For each time step, we apply Newton's method to the resulting nonlinear problem. The numerical results are presented for all three models in order to corroborate with the theoretical convergence order obtained.

中文翻译：

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具有边界条件的非线性波动方程的数值分析：狄利克雷、声学和不可穿透性


在本文中，我们提出了范数中的误差估计，涉及三个具有可变系数的波浪模型，并补充了初始条件和边界条件。前两个模型是具有狄利克雷、声学和非线性耗散不可渗透边界条件的非线性波动方程，而第三个模型是具有狄利克雷、声学和线性耗散不可渗透边界条件的线性波动方程。在数值分析领域，我们建立了两个关键定理来估计与每个模型相关的半离散和完全离散问题的误差。这些定理提供了空间和时间收敛速度的理论结果。为了进行数值模拟，我们在 Galerkin 方法的有限元空间中采用线性、二次和三次多项式基函数，并结合 Crank-Nicolson 方法进行时间离散化。对于每个时间步长，我们将牛顿法应用于所产生的非线性问题。给出了所有三个模型的数值结果，以证实所获得的理论收敛阶数。