We present a pollution-free first order system least squares (FOSLS) formulation for the Helmholtz equation, solved iteratively using a block preconditioner. This preconditioner consists of two components: one for the Schur complement, which corresponds to a preconditioner on $L_2(\Omega)$, and another defined on the test space, which we ensure remains Hermitian positive definite using subspace correction techniques. The proposed method is easy to implement and is directly applicable to general domains, including scattering problems. Numerical experiments demonstrate a linear dependence of the number of MINRES iterations on the wave number $\kappa$. We also introduce an approach to estimate algebraic errors which prevents unnecessary iterations.

翻译：本文提出了一种无污染的一阶系统最小二乘（FOSLS）公式来求解亥姆霍兹方程，并采用块预条件子进行迭代求解。该预条件子包含两个组成部分：一个用于处理对应于 $L_2(\Omega)$ 上预条件子的 Schur 补，另一个定义在测试空间上，我们通过子空间校正技术确保其保持 Hermitian 正定性。所提方法易于实现，并可直接应用于包括散射问题在内的一般域。数值实验表明，MINRES 迭代次数与波数 $\kappa$ 呈线性依赖关系。我们还引入了一种估计代数误差的方法，以防止不必要的迭代。