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In this paper we are concerned with numerical methods for nonhomogeneousHelmholtz equations in inhomogeneous media. We design a least squares methodfor discretization of the considered Helmholtz equations. In this method, anauxiliary unknown is introduced on the common interface of any two neighboringelements and a quadratic subject functional is defined by the jumps of thetraces of the solutions of local Helmholtz equations across all the commoninterfaces, where the local Helmholtz equations are defined on elements and areimposed Robin-type boundary conditions given by the auxiliary unknowns. Aminimization problem with the subject functional is proposed to determine theauxiliary unknowns. The resulting discrete system of the auxiliary unknowns isHermitian positive definite and so it can be solved by the PCG method. Undersome assumptions we show that the generated approximate solutions possessalmost the optimal error estimates with little "wave number pollution".Moreover, we construct a substructuring preconditioner for the discrete systemof the auxiliary unknowns. Numerical experiments show that the proposed methodsare very effective for the tested Helmholtz equations with large wave numbers.

A novel least squares method for Helmholtz equations with large wave numbers