Wave phenomenon is important in all areas of science and engineering. The Helmholtz equation arises from time-harmonic wave propagation, and the solutions are frequently required in many physical applications such as aero-acoustic, underwater acoustics, electromagnetic wave scattering, and geophysics. Even though computational methods have been successfully developed to solve many real and complex engineering problems, such as numerical simulations of complete aircraft and space-shuttle, it has been generally accepted that it is an extremely difficult task to solve the Helmholtz equation numerically, in particular, for the high-frequency cases.
In this talk, we discuss the difficulties associated with numerical solutions of Helmholtz equation for large wave numbers, namely the development of accurate discretization schemes, correct treatment of boundary conditions and reliable and robust computational methods for solving the resulting large system of indefinite and complex linear equations. We demonstrate that exact solutions for one-dimensional Helmholtz equation can be computed numerically if the rounding error is not considered. However, more works are needed to develop efficient and accurate computational techniques for two- and three-dimensional Helmholtz equations.