A polynomial collocation method for the numerical solution of weakly singular and singular integral equations on nonsmooth boundaries

We describe a numerical approach to solve (systems of) weakly singular and/or singular boundary integral equations defined on a piecewise smooth curve $\mathrm{\Gamma }=\bigcup {\mathrm{\Gamma }}_i$, i.e. with corners. It is well-known that these equations have solutions with a singular behaviour at the corners. In particular, we consider mixed boundary value problems (including Dirichlet and Neumann) for the 2D Laplace equation, which we reformulate as a system of boundary integral equations by using the single layer representation of the potential. Knowning a (smooth) parametrization of each arc ${\mathrm{\Gamma }}_i=\{a_i(t),t\in I_i\}$, our equations take the form: $$c_i{z}_i(t)+\sum _{j=1}^M\underset{I_j}{\int }{K}_{i,j}(s,t)z_j(s)ds=f_i(t)\$,\$t\in I_i\$,\$i=1,...,M$$ where the constant $c_i$ may be also zero. The functions $z_j(s)$ will be smooth in $I_j$, except possibly at the endpoints. Then, to smooth the endpoint singularities of the solutions $z_j$ we introduce a (nonlinear) smoothing transformation $\gamma$. Introducing this change of variable $\gamma$ and the Chebyshev weight function $\omega$ of the first kind (this last device is crucial to take advantage of some mapping properties of the corresponding weighted single layer operator) we proceed to solve numerically the a system of the type: $$c_i{v}_i(x)+\sum _{j=1}^M\underset{-1}{\overset{1}{\int }}{\bar{K}}_{i,j}(x,y)v_j(s)\omega (y)dy=g_i(x)\$,\$x\in [-1,1]\$,\$i=1,...,M$$ Now all the new unknowns $v_i$ are smooth functions, and we can approximate each one of them by a finite expansion of Chebyshev polynomials of first kind and apply a collocation method based on the zeros of the Chebyshev polynomials. This procedure is applied to several test problems. Our numerical approach has been extended also to corresponding 3D problems. In this last case, a very competitive cubature formulae, based on smoothing transformations too, have been proposed and used to evaluate surface integrals arising when discretizing boundary integral equations via the collocation method.
References

1. Monegato G., Scuderi L. - High order methods for weakly singular integral equations with non smooth input functions, Math. Comp. 67 (224), 1493-1515, 1998;
   
2. Monegato G., Scuderi L.- Global polynomial approximation for Symm's equation on polygons, Numer. Math. 86 (2000) 655-683.