# Why does standard finite elements method fails with transient problems?

Consider the one-dimensional wave equation $u_{tt} - c^2 u_{xx} = 0$ defined on a finite domain.

I have tried to simulate the propagation of a discontinuous wave by using FEM with piecewise-linear shape functions and time-stepping techniques such as Newmark scheme.

In the results, I always get the Gibbs phenomenon as I decrease the mesh size. I know that FEM assumes a solution that is separated in space and time, contrary to a wave where space and time are coupled. I want to know what is the mechanism that makes the solution so fail, since I would expect convergence.

• It would be helpful to say what equation exactly you are solving, and what your finite element basis-functions are. Used properly, time-domain finite element methods definitely are able to solve transient problems. – Wouter May 2 '18 at 15:04
• @Wouter I have edited the question. – cgyo May 5 '18 at 16:37