Splitting extrapolations for solving boundary integral equations of mixed boundary conditions on polygons by mechanical quadrature methods (Q1022841)

The author considers the problem: \[ \begin{aligned} \Delta u=0,\quad &\text{in}\quad\Omega \\ \alpha_{m}u+\beta_{m}\tfrac{\partial u}{\partial n}=g, \quad &\text{on}\quad \Gamma_{m}, \quad m=1,\dots,d, \end{aligned}\tag{*} \] where \(\Omega\subset\Re^{2}\) is a curved polygon with the edges \(\Gamma=\cup_{m=1}^{d}\Gamma_{m} .\) The differential problem is converted in an integral equation: \[ -\int_{\Gamma}\frac{\partial u(x)}{\partial n_{x}}\log|y-x|ds_{x}+\int_{\Gamma}u(x)\frac{\partial }{\partial n_{x}}\log|y-x|ds_{x}=\theta (y)u(y), \] where \(\theta (y)\) is a constant dependent on \(y \in \Gamma .\) The integral equation is then solved by a quadrature method dealing with singular integrals. If \(h_{m}\) is the mesh width of a curved edge \(\Gamma_{m}\) then the multivariate asymptotic expansions are found to be \(O(h^{3})\), \( h=maxh_{m}\) and by splitting extrapolation methods a convergence rate of \(O(h^{5})\) can be achieved. Numerical examples support the theoretical analysis.