On the boundary element method for billiards with corners

The boundary element method is one of the reliable numerical schemes to solve the eigenvalue problem of the Helmholtz equation, which is justified by the Fredholm theory for domains with a smooth boundary. When a domain has corners, however, the corresponding integral equation is singular, so that the boundary element method lacks its well-established base. Employing a cutoff technique, we here formulate a well-grounded version of the boundary element method, and also give a certain justification to the standard boundary element method even for domains with corners.