Abstract:

In this paper, the authors describe a fully discrete method for the numerical solution of the hyper-singular integral equation arising from the combined single and double layer potential approach for the solution of the exterior impedance boundary problem to the two-dimensional Helmholtz equation in smooth domains. The equation contains a hyper-singular operator. The authors develop an error analysis in a H\"{o}lder space setting with point-wise estimates and prove an exponential convergence rate for analytic boundaries and boundary data. Finally the numerical example is given to show the validity of the method.

Key words: Helmholtz equation, Impedance boundary condition, Integral equation, Hyper-singular operator