Boundary element preconditioners for a hypersingular integral equation on an interval
An almost optimal preconditioner for the iterative solution of the Galerkin equation for the hypersingular integral equation \[ (Du)(x)=-{1\over 2\pi} fp \int^1_{-1} {u(y)\over(x- y)^2} dy= f(x),\quad -1< x< 1, \] where \(fp\) means the finite part in the sense of Hadamard, are given. Let \[ (Vt)(x)= {1\over 2\pi} \int^1_{-1} \log\Biggl({1\over|x- y|}\Biggr) t(y) dy,\quad -1< x<1. \] Let \(W_h\) be any finite-dimensional space of functions defined on \([-1,1]\) such that \((Du_h, u_h)< \infty\) for all \(u_h\in W_h\). Let \(\{\phi_k\}^N_{k= 1}\) be a basis for \(W_h\). Let \[ D_h[l, k]= (D\phi_k, \phi_l),\;C_h[l,k]= (V^{- 1}\phi_k, \phi_l), \] \[ \widetilde C_h= M_h V^{-1}_h M_h,\;V_h[l, k]= (V\phi_k, \phi_l),\;M_h[l, k]= (\phi_k, \phi_l). \] Let a partition \(\Gamma_h\) consist of \(N+1\) subintervals determined by the points \(-1= x_0< x_1<\cdots< x_N< x_{N+ 1}= 1\), \(h_k= x_k- x_{k-1}\) \((1\leq k\leq N+ 1)\), \[ h_{\max}= \max_{1\leq k\leq N+1} h_k,\quad h_{\min}= \min_{1\leq k\leq N+1} h_k. \] For \(r\leq 2\) and \(0\leq m\leq r-2\), let \(S^{r, m}_0(\Gamma_h)\) denote the space of \(C^m\) splines of order \(r\), that vanish at \(\pm 1\). The main result of the paper is: Let \(\{\phi_k\}^N_{k= 1}\) be any basis for a trial space \(W_h\in\widetilde H^{1/2}(\Gamma)\), \(\Gamma= [-1,1]\) and let some other conditions fulfilled. Then \[ c_1(\widetilde C_h u,u)\leq (D_h u,u)\leq c_2\theta_h(\widetilde C_h u,u) \] for all \(u\in\mathbb{R}^N\), and hence \(k(\widetilde C^{- 1}_h D_h)\leq c\theta_h\), where \(k(D_h)\) is condition number of the operator \(D_h\). For \(W_h= S^{r, m}_0(\Gamma_h)\) the inequality \(\theta_h= [1+ \log h_{\min}]^2\) occur.
scientific article; zbMATH DE number 1421724