This proposal aims to develop novel numerical methods in porous media. Initially the Darcy model is examined under the framework of a Petrov-Galerkin enriched method (PGEM). The method consists in enriching differently trial and test funtions. The test function enrichments resembles residual-free bubbles, with zero condition for the normal compoenent of the velocity. This allows for static condensation and computation of the enrichment for the trial function. It turns out that this enrichment is a variation of the Raviart-Thomas. This is eliminated at the element level, so that the computational formulation of the current method is in fact that of the original spaces span by continuous piecewise linears and piecewise constants. A detailed study for the P1/P0 element and other higher order elements will be pursued along with their error analyses.
The methods developed in this work are key to simulate intricate large heterogeneuous domains which is a major challenge in porous media field, as in oil reservoir, contaminant transport and water resource problems, just to cite a few of them. The methods proposed herein contribute to decrease large scale computations since the matrices are no longer semi-definite. The methods are also competitive since the support of the unknows is equal to the original Galerkin method. In summary, we are pursuing more efficient methods that can overcome complex mixed methods which has dominated the field over many decades. The problems we are addressing have impact on environmental, energy and are of use to DOE.
