Convergence and Stability Analysis of Galerkin Finite Element Method for Telegraph Equation with Piecewise Continuous Arguments

Galerkin finite element method is presented to investigate the convergence and stability of a telegraph equation with piecewise continuous arguments in this paper. We derive the expression of analytic solution along with the conditions for its asymptotic stability by using the method of separation of variables. The convergence of semidiscrete case and fully discrete case is analyzed with [Formula: see text]-norm rigorously. Then the piecewise linear function is chosen as the bases of piecewise polynomial spline space, and we obtain the convergence order of semidiscrete case with [Formula: see text] and of fully discrete case with [Formula: see text], where [Formula: see text] and [Formula: see text] are the spatial step and temporal step, respectively. Moreover, the semidiscrete scheme is proved to be unconditionally stable and the conditions for stability of fully discrete scheme are derived under which the analytic solution is asymptotically stable. Finally, numerical examples are carried out to produce accuracy results which are acceptable and in good agreement with our theoretical results.