论文标题
非线性Helmholtz方程的高阶有限元方法
Higher-order finite element methods for the nonlinear Helmholtz equation
论文作者
论文摘要
在这项工作中，我们针对具有阻抗边界条件的非线性Helmholtz方程的任意但固定多项式度分析有限元方法。我们在波浪数$ k $，网格尺寸$ h $和多项式度$ p $之间的分辨率条件下显示了有限元解决方案的适当性和误差估计，形式为`$ k（kh）^{p+1} $足够小'''''''''''''''对于后者，我们证明了来自情况的$ h $中的对数依赖性$ p = 1 $ [H.〜WU，J.〜ZOU，\ emph {siam J.〜Numer。我们显示了两个不同的定点迭代方案的收敛性。数值实验说明了我们的理论结果，并比较了迭代方案与非线性和右侧数据的鲁棒性。
In this work, we analyze the finite element method with arbitrary but fixed polynomial degree for the nonlinear Helmholtz equation with impedance boundary conditions. We show well-posedness and error estimates of the finite element solution under a resolution condition between the wave number $k$, the mesh size $h$ and the polynomial degree $p$ of the form ``$k(kh)^{p+1}$ sufficiently small'' and a so-called smallness of the data assumption. For the latter, we prove that the logarithmic dependence in $h$ from the case $p=1$ in [H.~Wu, J.~Zou, \emph{SIAM J.~Numer.~Anal.} 56(3): 1338-1359, 2018] can be removed for $p\geq 2$. We show convergence of two different fixed-point iteration schemes. Numerical experiments illustrate our theoretical results and compare the robustness of the iteration schemes with respect to the size of the nonlinearity and the right-hand side data.
