Abstract: Since the exact solution of a boundary integral equation from BEM is unknown, the actual error in the approximate solution cannot be obtained. In this paper, it is shown that the residual error in the H 1/2 norm can be used as an estimate of error. The test problem is the boundary integral equation from the exterior Dirichlet problem for the Helmholtz equation with a smooth arc as its boundary, and it is solved using BEM both with and without the singular elements. The numerical results show that the use of the singular elements reduces the error significantly, and that the residual error in the H 0 norm approximates to one in the H 1/2 norm.