The Galerkin continuous finite element method for delay-differential equation with a variable term

In this paper, we introduce and analyze the continuous finite element method for a class of delay-differential equation with a variable term. At first for the case of the constant delay and based on an orthogonal expansion in an element we derive optimal superconvergence u − U = O(hp+2), p ⩾ 2 at the (p + 1)-order Lobatto points and u − U = O(h2p) at the integer nodal points. Next we prove the stability of linear and quadratic finite element schemes for the case of constant delay. For the case of variable delay, by use of the approximative to the continuous delay in every subinterval we decompose the original problem into a series of subproblems with constant delay. Finally we give a recursive schemes which can be compute by every subproblem.