Five-order Extrapolation Algorithms for Laplace Equation with Linear Boundary Condition

Laplace equation with linear boundary condition will be converted into a boundary integral equation(BIE) with logarithmic singularity following potential theory. In this paper, a Sidi quadrature formula is introduced to approximate the logarithmic singularity integral operator with O(h(3)) approximate accuracy order. A similar approximate equation is also constructed for the logarithmic singular operator, which is based on coarse grid with mesh width 2h. So an extrapolation algorithm is applied to approximate the logarithmic operator and the accuracy order is improved to O(h(5)). Moreover, the accuracy order is based on fine grid h. The convergence. and stability are proved based on Anselone's collective compact and asymptotic compact theory. Furthermore, an asymptotic expansion with odd powers of the errors is presented with convergence rate O(h(5)). Using h(5)-Richardson extrapolation algorithms(EAs), not only the approximation accuracy order can be improved to O(h(7)), but also an a posteriori error estimate can be obtained for constructing a self-adaptive algorithm, numerical examples are shown to verify its efficiency.