# Can Variational Inequalities handle non-symmetric matrices?

I am trying to enforce the discrete maximum principle (i.e., ensuring non-negative concentrations) for diffusion-type problems that have an anisotropic diffusivity tensor (e.g., tensor dispersion from velocity). For the standard diffusion equation, I could employ convex optimization since I will have a symmetric and positive definite matrix.

However, say I am working with the advection-diffusion equation. My problem is now non-symmetric and non-self-adjoint (and I have observed negative concentrations in its formulations), thus I cannot use convex optimization. Recently, I heard that PETSc has the Variational Inequality feature for its SNES solver, and I was told that this kind of solver is amenable for nonlinear problems. Now my question is, can this solver be applied to a problem like advection-diffusion?

It is indeed possible to write an obstacle problem for an advection-diffusion equation as a variational inequality: If $a(u,v)$ is the bilinear form corresponding to your advection-diffusion equation, the corresponding obstacle problem (in your specific case) is finding $u\geq 0$ such that $$a(u,v-u) \geq (f,v-u) \qquad \text{for all } v\geq 0.$$ In principle, no symmetry or convexity is required for this to be well-posed; however: