Journal of the
Korean Mathematical Society JKMS

A preconditioning algorithm is developed in this paper for the iterative solution of the linear system of equations resulting from the $p$-version boundary element approximation of the three dimensional integral equation with hypersingular operators. The preconditioner is derived by first making the nodal and side basis functions locally orthogonal to the element internal bases, and then by decoupling the nodal and side bases from the internal bases. Its implementation consists of solving a global problem on the wire-basket and a series of local problems defined on a single element. Moreover, the condition number of the preconditioned system is shown to be of order $O((1+\ln p)^7)$. This technique can be applied to discretization with triangular elements and with general basis functions.

Keywords: $p$-version boundary element method, hypersingular integral equation, iterative methods, preconditioning

MSC numbers: Primary 65N55, 65N38; Secondary 65F35