论文标题
基于ANSATZ的几何光学波波方法，用于具有可变波数的二维Helmholtz方程
A geometric optics ansatz-based plane wave method for two dimensional Helmholtz equations with variable wave numbers
论文作者
论文摘要
在本文中，我们开发了一种平面波类型方法，用于离散具有可变波数的同质Helmholtz方程。在提出的方法中，局部基础函数（在每个元素上）是由几何光学ANSATZ构造的，因此它们大致满足没有边界条件的同质Helmholtz方程。更准确地说，每个基础函数表示为指数平面波函数和多项式函数的乘积，其中指数函数中的相函数近似满足Eikonal方程，而多项式因子由与所考虑的Helmholtz方程相关的传输方程递归确定。我们证明，所得的平面波空间具有高阶$ h $ approximation作为标准平面波空间（仅可用于恒定波数的情况）。我们将提出的平面波空间应用于具有可变波数的非均匀Helmholtz方程的离散化，并建立了其有限元溶液的相应误差估计。我们报告了一些数值结果，以说明该方法的效率。
In this paper we develop a plane wave type method for discretization of homogeneous Helmholtz equations with variable wave numbers. In the proposed method, local basis functions (on each element) are constructed by the geometric optics ansatz such that they approximately satisfy a homogeneous Helmholtz equation without boundary condition. More precisely, each basis function is expressed as the product of an exponential plane wave function and a polynomial function, where the phase function in the exponential function approximately satisfies the eikonal equation and the polynomial factor is recursively determined by transport equations associated with the considered Helmholtz equation. We prove that the resulting plane wave spaces have high order $h$-approximations as the standard plane wave spaces (which are available only to the case with constant wave number). We apply the proposed plane wave spaces to the discretization of nonhomogeneous Helmholtz equations with variable wave numbers and establish the corresponding error estimates of their finite element solutions. We report some numerical results to illustrate the efficiency of the proposed method.
