Abstract: | This paper presents a new approach to verify the 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. It is also shown
that the vorticity under both steady and transient Darcy-Brinkman equations satisfy
maximum principles if the body force is conservative and the permeability is homogeneous
and isotropic. A discussion on the nature of vorticity under steady and transient
Darcy equations is also presented. Using several numerical examples, we will demonstrate
the predictive capabilities of the proposed a posteriori techniques in assessing the
accuracy of numerical solutions for a general class of problems, which could involve
complex domains and general computational grids. |