Petrov-Galerkin Enrichment Methods for Porous Media

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This proposal aims to develop novel numerical methods in porous media.Initially the Darcy model is examined under the framework of aPetrov-Galerkin enriched method (PGEM). The method consists inenriching differently trial and test funtions. The test functionenrichments resembles residual-free bubbles, with zero conditionfor the normal compoenent of the velocity. This allows forstatic condensation and computation of the enrichment for thetrial function. It turns out that this enrichment is a variationof the Raviart-Thomas. This is eliminated at the element level,so that the computational formulation of the current method isin fact that of the original spaces span by continuous piecewise linearsand piecewise constants. A detailed study for the P1/P0 elementand other higher order elements will be pursued along with theirerror analyses.The methods developed in this work are key to simulate intricatelarge heterogeneuous domains which is a major challenge in porousmedia field, as in oil reservoir, contaminant transport and waterresource problems, just to cite a few of them. The methods proposed hereincontribute to decrease large scale computations since the matricesare no longer semi-definite. The methods are also competitive sincethe 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, energyand are of use to DOE.