  This paper presents a new approach to verify accuracy of computational
simulations. We develop mathematical theorems which can serve as robust a
posteriori error estimation techniques to identify numerical pollution, check
the performance of adaptive meshes, and verify numerical solutions. We
demonstrate performance of this methodology on problems from flow thorough
porous media. However, one can extend it to other models. We construct
mathematical properties such that the solutions to Darcy and Darcy-Brinkman
equations satisfy them. The mathematical properties include the total minimum
mechanical power, minimum dissipation theorem, reciprocal relation, and maximum
principle for the vorticity. All the developed theorems have firm mechanical
bases and are independent of numerical methods. So, these can be utilized for
solution verification of finite element, finite volume, finite difference,
lattice Boltzmann methods and so forth. In particular, we show that, for a
given set of boundary conditions, Darcy velocity has the minimum total
mechanical power of all the kinematically admissible vector fields. We also
show that a similar result holds for Darcy-Brinkman velocity. We then show for
a conservative body force, the Darcy and Darcy-Brinkman velocities have the
minimum total dissipation among their respective kinematically admissible
vector fields. Using numerical examples, we show that the minimum dissipation
and total mechanical power theorems can be utilized to identify pollution
errors in numerical solutions. The solutions to Darcy and Darcy-Brinkman
equations are shown to satisfy a reciprocal relation, which has the potential
to identify errors in the numerical implementation of boundary conditions.
