  In this paper we show that we can use a modified version of the h-p spectral
element method proposed in \cite{duttora1,duttom,duttora2,tomarth} to solve
elliptic problems with general boundary conditions to exponential accuracy on
polygonal domains using nonconforming spectral element functions. A geometrical
mesh is used in a neighbourhood of the corners. With this mesh we seek a
solution which minimizes the sum of a weighted squared norm of the residuals in
the partial differential equation and the squared norm of the residuals in the
boundary conditions in fractional Sobolev spaces and enforce continuity by
adding a term which measures the jump in the function and its derivatives at
inter-element boundaries, in fractional Sobolev norms, to the functional being
minimized. In the neighbourhood of the corners, modified polar coordinates are
used and a global coordinate system elsewhere. A stability estimate is derived
for the functional which is minimized based on the regularity estimate in
\cite{babguo1}. We examine how to parallelize the method and show that the set
of common boundary values consists of the values of the function at the corners
of the polygonal domain. The method is faster than that proposed in
\cite{duttora1,duttom,tomarth} and the h-p finite element method and stronger
error estimates are obtained.
