On a Toeplitz representation of the \(H^{1/2}\) norm (Q1276297)

The authors present a new Toeplitz-type representation of the \(H^{1/2}\) norm in the space of linear finite elements defined on the (piecewise smooth) boundary of a bounded domain in \(\mathbb{R}^2\) which can be used as a Schur complement preconditioner in finite element and boundary element methods. The proposed norm representation is a dense matrix, but contains only \(O(\log n)\) different entries at each row, where \(n\) is the number of rows. So a matrix-vector computation can be done by \(O(n \log n)\) arithmetric operations without using the fast Fourier transform, leading to a simpler implementation of the preconditioned conjugate gradient method. Some numerical results are also given.