In this paper we are concerned with numerical methods for nonhomogeneous Helmholtz equations in inhomogeneous media. We design a least squares method for discretization of the considered Helmholtz equations. In this method, an auxiliary unknown is introduced on the common interface of any two neighboring elements and a quadratic subject functional is defined by the jumps of the traces of the solutions of local Helmholtz equations across all the common interfaces, where the local Helmholtz equations are defined on elements and are imposed Robin-type boundary conditions given by the auxiliary unknowns. A minimization problem with the subject functional is proposed to determine the auxiliary unknowns. The resulting discrete system of the auxiliary unknowns is Hermitian positive definite and so it can be solved by the PCG method. Under some assumptions we show that the generated approximate solutions possess almost the optimal error estimates with little "wave number pollution". Moreover, we construct a substructuring preconditioner for the discrete system of the auxiliary unknowns. Numerical experiments show that the proposed methods are very effective for the tested Helmholtz equations with large wave numbers.

中文翻译：

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大波数亥姆霍兹方程的一种新的最小二乘法

在本文中，我们关注非齐次介质中非齐次亥姆霍兹方程的数值方法。我们设计了一种最小二乘法来离散化所考虑的亥姆霍兹方程。在该方法中，在任意两个相邻元素的公共界面上引入辅助未知数，并通过所有公共界面上的局部亥姆霍兹方程解的迹的跳跃定义二次主体泛函，其中局部亥姆霍兹方程为定义在元素上，并强加了由辅助未知数给出的罗宾型边界条件。提出了主题泛函的最小化问题来确定辅助未知数。由此产生的辅助未知量离散系统是厄米正定的，因此可以用 PCG 方法求解。在一些假设下，我们表明生成的近似解几乎具有最佳误差估计，而“波数污染”很少。此外，我们为辅助未知数的离散系统构建了一个子结构预处理器。数值实验表明，所提出的方法对于具有大波数的测试亥姆霍兹方程非常有效。