A qualocation method for parabolic partial differential equations

In this paper a qualocation method is analysed for parabolic partial differential equations in one space dimension. This method may be described as a discrete H¹-Galerkin method in which the discretization is achieved by approximating the integrals by a composite Gauss quadrature rule. An O (h^4-i) rate of convergence in the W^i.p norm for i = 0, 1 and 1 ⩽ p ⩽ ∞ is derived for a semidiscrete scheme without any quasi-uniformity assumption on the finite element mesh. Further, an optimal error estimate in the H² norm is also proved. Finally, the linearized backward Euler method and extrapolated Crank-Nicolson scheme are examined and analysed.