The scaled boundary finite-element method is a new semi-analytical approach to computational mechanics developed by Wolf and Song. The method weakens the governing differential equations by introducing shape functions along the circumferential coordinate direction(s). The weakened set of ordinary differential equations is then solved analytically in the radial direction. The resulting approximation satisfies the governing differential equations very closely in the radial direction, and in a finite-element sense in the circumferential direction. This paper develops a meshless method for determining the shape functions in the circumferential direction based on the local Petrov-Galerkin approach. Increased smoothness and continuity of the shape functions is obtained, and the solution is shown to converge significantly faster than conventional scaled boundary finite elements when a comparable number of degrees of freedom are used. No stress recovery process is necessary, as sufficiently accurate stresses are obtained directly from the derivatives of the displacement field.