# Are there discontinous solutions of the wave equation in two dimensions?

Can you construct a discontinous solution for the wave equation $$u_{tt} - c^2 \Delta u = f$$ with homogenous dirichlet boundary conditions on the domain $$\Omega = [0,1]^2$$?

Background: I programmed a tool which solves the wave equation in 1d ($$u_{tt} - c^2 u_{xx} = f$$, $$\Omega = [0,1]$$) and 2d ($$u_{tt} - c^2 ( u_{xx} + u_{yy}) = f$$, $$\Omega = [0,1]^2$$) with dirichlet boundary conditions and a starting deflection $$u_0$$ and and a starting velocity $$v_0$$. I use the finite element in space and leapfrog/crank-nicolson/hht-$$\alpha$$ in time. Therefore I expect a convergence of $$\mathcal{O}(h^2 + \Delta t^2)$$ if $$u\in C^4$$. In one dimension, I use d'Alembert to calculate the reference solution. This also works with discontinous solutions. I want to test the convergence speed in 2d if $$u \notin C^4$$, for example if $$u$$ is discontinous. Can I construct a reference solution in 2d to calculate the error ($$L_2$$-norm)?

• Have you tried discontinuous initial values? – daw Feb 22 at 9:20
• I assume you mean weak solutions? – Klaus Feb 22 at 9:25
• Yes, I'm looking for weak solutions. I'm not quite sure what you mean with discontinous inital values. If I set (for example) $u= cos(t) * some indicator function$ I would get a discontinous inital condition, but then I have problems to find the right side because of the missing derivatives of this solution – Julian H Feb 22 at 13:09