The Finite Element Methods For The Delay-differential Equations

For a class of delay-differential equations, the linear and nonlinear delay-equation with one-delay term and multi-delay terms, this paper systemic studies the finite element methods and their superconvergence. For the type of the finite element methods, we study the continuous finite element and the discontinuous finite element.The following is an outline of the main results of the paper:1. By application of adjustment orthogonal approximate technique in an element, superconvergence of continuous finite elements for linear delay-differential equations with one variabe delay term is studied and ultracon-vergence of its finite element derivative recovery is deduced. Then we derive superconvergence of continuous finite element for the linear delay-differential equations with multi-delay terms by the mathematics induction. The above theoretical results are tested by three numerical examples.2. For the nonlinear delay-differential equations, firstly discuss super-convergence of continuous finite element methods for the case with one-delay term. Next expand the results to the case with multi-delay terms. Finally the theoretical results are tested by two numerical examples.3. For the delay-differential equations with one-delay term, first we study superconvergence of discontinuous finite element methods for the linear case. Next we expand the results to the nonlinear case. Finally the theoretical results are tested by two examples of the linear and nonlinear delay-differential equations.4. Finally for the two order delay-differential equations, we simply discuss the continuous finite element approximative scheme and the discontinuous finite element approximative scheme.