Note on a New High Order Piecewise Linear Finite Element Approximation for the Wave Equation in One Dimensional Space

Abstract

We consider the piecewise linear finite element method in space for solving the one dimensional wave equation on general spatial meshes. The discretization in time is performed using the Newmark method. We show that the error between the finite element approximate solution and the piecewise linear interpolant of the exact solution is of order $(h+k)^2$ in several discrete norms. We construct a new approximation of order $(h+k)^3$. This new third order approximation can be computed using the same linear systems used to compute the finite element approximate solution with the same matrices while the right hand sides are changed. The matrices used to compute this new high-order approximation are tridiagonal and consequently the systems involving these matrices are easily to solve. The convergence analysis is performed in several discrete norms.