Mechanical quadrature methods and their extrapolations for solving boundary integral equations of the conductivity problem with discontinuous media

Mixed boundary value problems of the conductivity equation ∇.(γ∇u)=0∇.(γ∇u)=0 with discontinuous media are converted into boundary integral equations by using the single layer potential. For solving the induced boundary integral equations, a mechanical quadrature method is proposed for periodic Fredholm integral equations with logarithmic singularity, which possesses a high order accuracy O(h3)O(h3), less computational complexity and asymptotic expansion of the errors. By means of Richardson extrapolation, an approximation with a higher accuracy order O(h5)O(h5) is obtained. Moreover, a posteriori error estimate for the algorithm is derived, which can be used to constructed adaptive algorithm. Several numerical examples show that the accuracy order of approximation is very high, the extrapolation and a posteriori error estimates are also very effective.