# How to avoid negative values of numerical solution of transport equation using FEM scheme?

The transport equation is actually an advection-diffussion-reaction equation, which has the form as

$$\frac{\partial C}{\partial t} + v_1 \frac{\partial C}{\partial x} + v_2 \frac{\partial C}{\partial x} = D \left(\frac{\partial^2C}{\partial x^2}+\frac{\partial^2C}{\partial y^2}\right)-R(x,y)\cdot C+S$$

where $C$ is the unknown substrate concentration, $v_1$ and $v_2$ are the fluid velocities in the $x$ and $y$ direction, respectively, $D$ is the diffusion coefficient, $R$ is the reaction term, and $S$ is the source term. I use the traditional FEM scheme with backward Euler for the time advancing. But it seems that there are some negative values appearing in the numerical solution I solved. Is there any method/scheme to avoid negativeness?