AI 中文总结
研究一般有界多边形/多面体域上非线性波动方程全离散格式，结合指数欧拉时间积分器与有限元空间离散化，证明低正则性解误差估计，表明高阶有限元方法优势，数值实验证实收敛行为。
AI 中文摘要
我们研究了一般有界多边形/多面体域上非线性波动方程的全离散格式，其初始数据\((u^0,v^0)\in H^\gamma(\Omega)\times H^{\gamma - 1}(\Omega)\)，\(0\lt\gamma\leq1\)，并满足与齐次狄利克雷边界条件相关的自然相容性条件。该格式将指数欧拉时间积分器与有限元空间离散化相结合。与大多基于傅里叶谱离散化的现有低正则性误差分析不同，我们的方法适用于一般有界域和有限元空间离散化。我们证明了低正则性解的严格误差估计。分析基于底层椭圆算子的频率分解，仅用作解析正则化工具而非实际实现中使用，这使得低正则性技术能扩展到傅里叶谱框架之外。结果还表明，即使对于索伯列夫正则性有限的解，高阶有限元方法在空间逼近中仍具优势。不同域和不同多项式次数的数值实验证实了预测的收敛行为。
英文摘要
We study a fully discrete scheme for nonlinear wave equations on general bounded polygonal/polyhedral domains with initial data $(u^0,v^0)\in H^γ(Ω)\times H^{γ-1}(Ω)$, $0<γ\le 1$, subject to the natural compatibility condition associated with the homogeneous Dirichlet boundary condition. The scheme combines an exponential Euler time integrator with a finite element spatial discretization. In contrast to existing low-regularity error analyses, which are mostly based on Fourier spectral discretizations, our approach applies to general bounded domains and finite element spatial discretizations. We prove rigorous error estimates for low-regularity solutions. The analysis is based on a frequency decomposition of the underlying elliptic operator, used solely as an analytical regularization device and not in the actual implementation, which allows low-regularity techniques to be extended beyond the Fourier spectral framework. The results also indicate that higher-order finite element methods remain advantageous in spatial approximation even for solutions of limited Sobolev regularity. Numerical experiments on different domains and with different polynomial degrees confirm the predicted convergence behavior.
Comments32 pages, 7 figures
