On mechanics-based a posteriori criteria to assess accuracy of numerical solutions for Darcy and Darcy-Brinkman equations

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. 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.