On the performance of maximum-principle enforcing methods applied to large-scale subsurface problems

It is well known that numerical formulations (either finite element, finite volume or finite difference) do not meet maximum principles and the non-negative constraint for anisotropic diffusion equations. But these mathematical properties and physical constraints are important for predictive simulations in subsurface modeling. Recently, optimization-based methodologies have been proposed for diffusion-type equations that respect maximum principles and meet the non-negative constraint on general computational grids. Till date these methodologies have been tested only on small-scale academic problems with few thousands of degrees-of-freedom. But for practical problems in subsurface modeling, the degrees-of-freedom easily run into millions and sometimes into billions. The purpose of this research is to systematically study the performance of the non-negative methodologies for large-scale problems and in a parallel setting. We shall use PETSc for parallel environment, and TAO for parallel optimization solvers. Numerical simulations on real sites using our methodologies will be presented.