In this paper a qualocation method is analysed for parabolic partial differ
ential equations in one space dimension. This method may be described as a
discrete HI-Galerkin method in which the discretization is achieved by appr
oximating the integrals by a composite Gauss quadrature rule. An O(h(4-i))
rate of convergence in the W-i,W-p norm for i = 0, 1 and 1 less than or equ
al to p less than or equal to infinity is derived for a semidiscrete scheme
without any quasi-uniformity assumption on the finite element mesh. Furthe
r, an optimal error estimate in the H-2 norm is also proved. Finally, the l
inearized backward Euler method and extrapolated Crank-Nicolson scheme are
examined and analysed.