Of Helmholtz Equation Finite Element Method

Helmholtz equation usually appears in the area of acoustics, electromagnetics and elastic mechanics. It is very important in science research. A lot of literatures have made theoretic analysis of numerical method about the equation, but the research in solving the equation is comparatively less. In this paper, we absorb some conclusions mentioned in other literatures and make some analysis in theory, but place importance on the numeric experiment of the equation by finite element method. We use linear interpolation, Hermite interpolation and triangular interpolation to make the Shape-Function for one-dimension problem. Then we give the error data for the numeric solution by numeric experiment.We use triangle element and rectangle element to decompose the two-dimension area. Then we make the Shape-Function on the area by linear interpolation, Hermite interpolation and triangular interpolation. We also get some base functions from the analytics solution of the homogeneous Helmholtz equation, and establish the Shape-Function by them. Then we give the error data for the numeric solution by numeric experiment.We use triangular prism element and cuboid element to decompose the three-dimension area. Then we make the Shape-Function on the area by linear interpolation, Hermite interpolation and triangular interpolation. Then we give the error data for the numeric solution by numeric experiment.We also depict the picture of error data for both one-dimension problem and two-dimension problem, from which we can see the error spreaded in the defined area.By plenty of numerical experiments, we prove that triangular Shape-Function can get a good precision, especially for the periodic problems. it is also suitable to solve the equation with large wave number.