Mechanics-Based Solution Verification for Porous Media Models (Communications in Computational Physics, 2020)
This paper presents a new approach to verify the accuracy of computationalsimulations. We develop mathematical theorems which can serve as robust a posteriorierror estimation techniques to identify numerical pollution, check the performance ofadaptive meshes, and verify numerical solutions. We demonstrate performance of thismethodology on problems from flow thorough porous media. However, one can extendit to other models. We construct mathematical properties such that the solutionsto Darcy and Darcy-Brinkman equations satisfy them. The mathematical propertiesinclude the total minimum mechanical power, minimum dissipation theorem, reciprocalrelation, and maximum principle for the vorticity. All the developed theoremshave firm mechanical bases and are independent of numerical methods. So, these canbe 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 ofboundary conditions, Darcy velocity has the minimum total mechanical power of allthe kinematically admissible vector fields. We also show that a similar result holds forDarcy-Brinkman velocity. We then show for a conservative body force, the Darcy andDarcy-Brinkman velocities have the minimum total dissipation among their respectivekinematically admissible vector fields. Using numerical examples, we show that theminimum dissipation and total mechanical power theorems can be utilized to identifypollution errors in numerical solutions. The solutions to Darcy and Darcy-Brinkmanequations are shown to satisfy a reciprocal relation, which has the potential to identifyerrors in the numerical implementation of boundary conditions. It is also shownthat the vorticity under both steady and transient Darcy-Brinkman equations satisfymaximum principles if the body force is conservative and the permeability is homogeneousand isotropic. A discussion on the nature of vorticity under steady and transientDarcy equations is also presented. Using several numerical examples, we will demonstratethe predictive capabilities of the proposed a posteriori techniques in assessing theaccuracy of numerical solutions for a general class of problems, which could involvecomplex domains and general computational grids.