The Hp Type Galerkin Time Step Method For Second-order Nonlinear Delay Differential Equations

In this paper,we mainly study the h-p version Galerkin time stepping methods for non-linear second-order delay differential equations.Firstly,we rewrite the original equation into a first-order system,and then we propose an h-p version of the discontinuous Galerkin method for the nonlinear second-order delay differential equations.We derive a-priori error estimates in the L~2-and L~?-norms that are fully explicit in the local time steps,the local polynomial degrees,and the local regularity of the exact solution.Secondly,we further develope an h-p version of the C~0-continuous Petrov-Galerkin method for the nonlinear second-order delay differential equations.We also obtain a-priori error estimate in the H~1-norm that is totally explicit eith respect to the local discretization and regularity parameters of the solution.In particular,for problems with singular solutions,we prove that both methods can reach exponential rates of convergence by using geometrically refined time steps combined with linearly increasing approximation orders.Finally,We verify the theoretical results through a series of numerical examples.