Finite Element Method for the One-Dimensional Telegraph Equation

Abstract

A linear telegraph equation with periodic boundary conditions is chosen for a model
problem in this paper. The model problem is used in order to be illustrated a new finite element
method for solving hyperbolic boundary value problems. There are various weak forms of the
telegraph equation that have been in use up to now. An original weak problem related to the
problem of interest is obtained. A detailed multigrid algorithm describing the essence of the
considered method is developed. Numerical simulations including smoothing of an error
function and a graph of an approximate solution are demonstrated. An approximate asymptotic
rate of convergence is calculated by applying three successive triangulations and cubic trial
functions.