# Do DG methods for the Helmholtz equation always return positive quantities?

Helmholtz Diffusion equation with reaction term: $$k\Delta u + u = f ~ \text{in} ~\Omega$$ $$\nabla u \cdot \mathbf{n} = 0 ~ \text{in} ~\partial \Omega$$

For sufficiently small $k$ (relative to the mesh), there can be oscillations (with Lagrange elements) that return a negative $u$ in some regions, specially when $f$ is a step function. I tried to apply a DG method with the Symmetric Interior Penalty formulation, and I am still seeing these negative values. Is this possible to happen? I was told that DG methods preserve positivity and cannot return negative values. Is there any way to preserve positivity in DG methods for elliptic equations?

Preserve positivity: If $f(x)>0 ~\forall x\in\Omega$, then $u(x)>0 ~\forall x\in\Omega$

Edit again: What if the boundary conditions are of zero flux? Edited the problem.

• What do you mean by preserving positivity? Do you mean that $u(x) > 0\ \forall x \in \Omega$? – nicoguaro Jul 20 '17 at 14:32
• Yes, I edited my question for clarity. – balborian Jul 20 '17 at 15:15
• I'm not sure that is correct. If we take the 1D version of your equation $k \frac{d^2u}{dx^2} + u =f$, and take $f=k=1$, the solution is: $u(x) = A \sin(x) + B \cos(x) + 1$. This is positive for some values of $A$ and $B$, but not always. – nicoguaro Jul 20 '17 at 16:21
• Correct -- there is no reason why the solution should be positive. In fact, for particular values of $k$ (the inverse of the eigenvalues of the Laplacian), the equation is singular and there is no unique solution any more. – Wolfgang Bangerth Jul 20 '17 at 20:50
• Does the situation change if there is zero flux in the boundary? – balborian Jul 22 '17 at 17:19