A Posteriori Error Estimation and Up-Scaling for Mixed Finite Element Methods

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9707015 Todd Arbogast A POSTERIORI ERROR ESTIMATION AND UP-SCALING FOR MIXED FINITE ELEMENT METHODS This project concerns the approximation of second order elliptic and parabolic partial differential equations by mixed finite element methods on logically rectangular grids. The first objective is to develop a local a posteriori error estimator or indicator, so that spatial errors can be localized. We propose to estimate the error by exploiting the equivalence between mixed and non-conforming methods for rectangular elements. The second objective is to develop up-scaling or homogenization techniques for highly variable coefficients and point-like sources, i.e., for resolving fine length scales in the model system that are below the size of a practical computational mesh. The error associated with using these up-scaling techniques must be quantifiable. We propose to base our techniques on the discrete equations, and use the coarse scale Raviart-Thomas projection operator which preserves the flux across any element face. The third objective is to demonstrate the applicability of the techniques in a practical setting; we consider the simulation of subsurface flow. The first two objectives are complementary. Error estimation would allow us to refine the mesh where the solution is ill behaved, as near sharp fronts, local heterogeneity, or sources (i.e., wells), so that computational effort can be concentrated to resolve the major length scales in both the data and the solution. Up-scaling would allow us to further resolve some scales below the mesh size. Our understanding of fluid flow underground is important to a range of activities, including the clean-up of ground-water contamination and oil and gas production. Ground-water supplies are increasingly threatened by contaminants introduced into the environment by improper disposal or accidental release. U.S. petroleum production has declined markedly in recent years. These problems can be ameliorated by complex engineering processes that require careful design and monitoring, which in turn depend on our ability to simulate on a computer the movement of fluids underground. Our computer simulations must be sufficiently detailed that we can further predict physical, chemical, and biological processes and the consequences of human intervention. Such simulation requires that we approximate accurately the differential equations governing the movement and interaction of the fluids. It is difficult to do this for a number of reasons, but the most basic is a lack of resolution: we can only use data and compute fluid velocities at a small number of grid points in space. To date there has been no effective way to estimate the error in the approximation to the fluid velocity. If we knew that our errors were large, we could take corrective action by increasing the grid resolution, up to the limits of the computer resources available. Beyond that limit, it is necessary to approximate certain very small-scale quantities below the grid scale (such as local variations in rock properties) by replacing them by some appropriately defined average quantities. We address these concerns in this proposal, and demonstrate the applicability of our techniques in practical settings to gain their acceptance by engineers.