AI 中文总结
研究光滑有界区域上带动态边界条件的非线性波动方程，结合\(k\)次等参体-面有限元与指数积分器，证明其全离散逼近在低正则性下收敛，给出时空收敛阶，通过弱范数框架解决非协调几何问题，数值实验验证结果。
AI 中文摘要
我们研究了在光滑有界区域上具有动态边界条件的非线性波动方程，并在低正则性条件下分析了一种全离散逼近方法。该方法将\(k\)次等参体-面有限元与时间上的指数积分器相结合。仅假设精确解的能量有界，我们证明了位移-速度对在弱范数\(L^2(\Omega;\Gamma)\times H^{-1}(\Omega;\Gamma)\)下的收敛性。该格式在时间上达到一阶收敛，在空间上，当\(k = 1\)时收敛阶为\(h^{2/3}\)，当\(k\geq2\)时收敛阶为\(h^{(k + 2)/(k + 3)}\)。特别地，这些速率表明即使在低正则性下高阶有限元仍保持可证明的渐近优势。一个核心困难是连续和离散的体-面问题是在不同几何上提出的，因此必须在弱范数下直接比较。为了解决这个问题，我们基于提升和伴随提升算子为非协调几何开发了一个弱范数框架，并结合频率分解论证。据我们所知，这是在非协调体-面有限元设置下具有动态边界条件的非线性波动方程的第一个全离散低正则性收敛结果。数值实验证实了预测的速率，并说明了高阶方法的提高效率。
英文摘要
We study nonlinear wave equations with dynamic boundary conditions on smooth bounded domains and analyze a fully discrete approximation in the low-regularity regime. The method combines isoparametric bulk--surface finite elements of degree $k$ with an exponential integrator in time. Assuming only bounded energy of the exact solution, we prove convergence of the displacement--velocity pair in the weak norm $L^2(Ω;Γ)\times H^{-1}(Ω;Γ)$. The scheme achieves first-order convergence in time and spatial convergence of order $h^{2/3}$ for $k=1$ and $h^{(k+2)/(k+3)}$ for $k\ge 2$. In particular, these rates show that higher-order finite elements retain a provable asymptotic advantage even at low regularity. A central difficulty is that the continuous and discrete bulk--surface problems are posed on different geometries and must therefore be compared directly in weak norms. To address this, we develop a weak-norm framework for non-conforming geometries based on lift and adjoint-lift operators, combined with a frequency-decomposition argument. To the best of our knowledge, this is the first fully discrete low-regularity convergence result for nonlinear wave equations with dynamic boundary conditions in a non-conforming bulk--surface finite element setting. Numerical experiments confirm the predicted rates and illustrate the improved efficiency of higher-order methods.
Comments45 pages, 5 figures
