# How would one solve the wave equation using the finite element method?

I would like to solve $$\alpha u_{tt} = -\nabla^2u$$ with $\frac{\partial u}{\partial n} = 1$. On using a Galerkin approximation I obtain $$M\ddot{c}=\frac{1}{\alpha}(Dc+b)$$ where $M$ is the mass matrix and and $D$ is the stiffness matrix. I note that I can write this in the form $$\dot{U} = ZU+B$$

where $U = [c, \dot{c}]^T$ and $Z$ has off diagonal blocks top right: $I$ and bottom left $M^{-1}D$ and $B$ has only non-trivial entries in the "second half" of the vector given by $M^{-1}b$.

We can go ahead and use in implicit finite difference scheme on this problem then.(?)

I know my explanation is sparse but I am hoping that someone who might have solved this before can confirm whether this is a viable method. My solution is very "unstable" looking.

Perhaps I am being really naive using this method. Are there simple methods to solve this problem?

• Do you have initial conditions? What is the geometry of your domain? – nicoguaro Mar 10 '18 at 22:51
• The formulation you use is exactly the one I'm using here: dealii.org/8.5.0/doxygen/deal.II/step_23.html – Wolfgang Bangerth Mar 11 '18 at 5:59
• I am trying both a disk and a square. I am using Neumann boundary conditions with constant initial displacement and zero initial velocity(hence should remain constant). I think there must be something wrong with my implementation. Thanks for the link Wolfgang. – ZingyMcGhee Mar 11 '18 at 7:59
• Thanks, it all worked out. It was the classic case of missing minus sign. – ZingyMcGhee Mar 13 '18 at 12:29
• Great! You can answer your own question and accept your answer. It is considered good citizenship at SO. – Anton Menshov Mar 17 '18 at 2:04