Abstract: We considered  grating problems of random interface. Firstly, it was abstracted to be Helmholtz equations with relevant boundary conditions, and given  the perturbation interface. Secondly, a numerical method based on shape derivative and the finite element method was proposed  to solve the grating problem of random interface. The expected second\|order approximation and the third-order approximation of variance for the grating problem of random interface were obtained. Finally, the estimation of error and numerical examples were given.

Key words: Helmholtz equation, finite element method, shape derivative, random interface