二阶谐波产生散射的相容有限元方法分析
Analysis of a Conforming Finite Element Method for Second-Harmonic Generation Scattering
AI总结：
本文分析了采用精确狄利克雷-纽曼边界条件的二阶谐波产生散射问题的相容有限元方法，证明了解的存在唯一性，推导了误差估计并验证了方法的收敛性。
AI中文摘要：
在频域中，非线性声学与电磁波传播在特定条件下可由包含两个耦合非线性亥姆霍兹方程的二阶谐波产生系统建模。本文对采用精确狄利克雷-纽曼边界条件截断的标量二阶谐波产生散射问题的相容有限元近似展开分析，离散化使用次数为$p$的连续分段多项式有限元。在涉及入射场与非线性 susceptibility 的小数据条件下，我们证明了规定小球内连续与离散非线性解的存在性与唯一性；在同一条件下，通过结合线性伽辽金拟最优性与耦合非线性项小数据稳定性估计的非线性塞型论证，推导了拟最优的先验$H^1$误差估计。我们还分析了用于计算离散非线性解的不动点迭代的收敛性，量化了有限元近似与非线性迭代误差的综合效应。一个制造解实验验证了预测的有限元收敛速率，额外的基于PML的散射计算则展示了非线性不动点求解器在二维与三维中的表现。
英文摘要：
In the frequency domain, nonlinear acoustic and electromagnetic wave propagation can, in certain regimes, be modeled by second-harmonic generation systems consisting of two coupled nonlinear Helmholtz equations. We analyze a conforming finite element approximation of a scalar second-harmonic generation scattering problem truncated using an exact Dirichlet-to-Neumann boundary condition. The discretization uses continuous piecewise polynomial finite elements of degree $p$. Under small-data conditions involving the incident-field and nonlinear susceptibilities, we prove existence and uniqueness of the continuous and discrete nonlinear solutions in a prescribed small ball. In the same regime, we derive quasi-optimal a priori $H^1$-error estimates by a nonlinear Céa-type argument combining linear Galerkin quasi-optimality with small-data stability estimates for the coupled nonlinearities. We also analyze the convergence of the fixed-point iteration used to compute the discrete nonlinear solution and quantify the combined effects of finite element approximation and nonlinear iteration error. A manufactured-solution experiment illustrates the predicted finite element convergence rates, while additional PML-based scattering computations demonstrate the behavior of the nonlinear fixed-point solver in two and three dimensions.
