A preconditioner, based on a two-level mesh and a two-level orthogonal
ization, is proposed for the h-p version of the finite element method
for two dimensional elliptic problems in polygonal domains. Its implem
entation is in parallel on the subdomain level for the linear or bilin
ear (nodal) modes, and in parallel on the element level for the high o
rder (side and internal) modes. The condition number of the preconditi
oned linear system is of order [GRAPHICS] where H-i is the diameter of
the i-th subdomain, h(i) and p(i) are the diameter of elements and th
e maximum polynomial degree used in the subdomain. This result reduces
to well-known results for the h-version (i.e. p(i) = 1) and the p-ver
sion (i.e. h(i) = H-i) as the special cases of the h-p version.