For the scattering problems of acoustic wave for an open arc in two dimensions, we give a uniqueness and existence analysis via the single layer potential approach leading to a system of integral equations that contains a weakly singular operator. For its numerical solutions, we describe an $O(h^{3})$
O
(
h
3
)
order quadrature method based on the specific integral formula including convergence and stability analysis. Moreover, the asymptotic expansion of errors with odd power $O(h^{3})$
O
(
h
3
)
is got and the Richardson extrapolation algorithm (EA) is used to improve the accuracy of numerical solutions. The efficiency of the method is illustrated by a numerical example.
In this paper, we derive the convergence for the high-accuracy algorithm in solving the Dirichlet problem of the modified Helmholtz equation. By the boundary element method, we transform the system to be a boundary integral equation. The high-accuracy algorithm using the specific quadrature rule is developed to deal with weakly singular integrals. The convergence of the algorithm is proved based on Anselone’s collective compact theory. Moreover, an asymptotic error expansion shows that the algorithm is of order Oh03. The numerical examples support the theoretical analysis.
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