On the approximate solution of elliptic pseudodifferential equations on a smooth closed curve

Abstract

We present a procedure for the numerical treatment of elliptic pseudodifferential equations on smooth closed curves using the parametrix for the corresponding pseudodifferential operators combined with the trigonometric collocation procedure. This requires $O(N\mathrm{In}N)$ significant operations to find $N$ coefficients of the trigonometric polynomial which approximates the solution. We prove that the asymptotic error estimate obtained, including the quadrature errors, is of optimal order in the scale of Sobolev spaces.