A qualocation method for parabolic partial differential equations

Abstract

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 HI-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,W-p norm for i = 0, 1 and 1 less than or equal to p less than or equal to infinity is derived for a semidiscrete scheme without any quasi-uniformity assumption on the finite element mesh. Further, an optimal error estimate in the H-2 norm is also proved. Finally, the linearized backward Euler method and extrapolated Crank-Nicolson scheme are examined and analysed.