The investigator and his colleague Bernardo Cockburn, in a collaborative project, develop adaptive numerical methods for problems with moving interfaces and long-time dynamics. These problems incorporate multiple temporal and spatial scales. Finite element methods have long been used for solving partial differential equations. Recently, methods using discontinuous approximating spaces have become popular, especially for nonsteady convection-diffusion problems. Two such methods are the local discontinuous Galerkin method (LDG) and the Godunov-mixed method (GMM), developed by the investigators and their collaborators. These methods have the advantage that they are based on local conservation and approximate shocks and sharp gradients with no spurious oscillations, which is important in many convection-diffusion applications. They also lend themselves to parallel computation. Both methods have been implemented computationally, and a priori error estimates have been derived; however, no adaptive strategies based on these methods have been developed. It has long been recognized that adapting the finite element mesh and time-step during a simulation is desirable for obtaining an accurate solution. In order to successfully adapt the mesh to guarantee that the actual error is below a given tolerance, it is essential to develop a posteriori error estimates that measure the actual error as a funtion of the mesh, time step and the computed solution. While a substantial literature exists for such estimates for steady problems and conforming finite element spaces, little research has been done for nonsteady problems and discontinuous methods. In this project, the investigators and their colleagues develop a posteriori estimates for the LDG and GMM methods for convection-diffusion equations, with emphasis on three important applications: shallow water flow, chemically reactive transport in porous media and surface water, and the modeling of brain tumor cell growth and treatment. The basis for these estimates is the so-called approximate adjoint equation methodology; however, other ad-hoc methods which may potentially be more efficient are also investigated. These estimates are novel for the applications as well as the numerical methods.
The applications of interest are important to industry, government laboratories and departments, and state agencies. Modeling of flow patterns in shallow water systems (e.g. bays and estuaries) is important for understanding, for instance, the environmental impacts of oil spills and the economic impacts of dredging, and can also be useful in tracking storm surges during hurricanes and other extreme weather events. Modeling of transport of chemical species in groundwater and surface water is important for understanding waste disposal and pollution remediation. The modeling of brain tumors can be useful in predicting tumor growth and examining potential treatments. These applications, though varied, share common mathematical characteristics and can utilize similar numerical simulation methodologies. Lacking for these applications and methodologies are sound, mathematically based tools for controlling and adapting the simulations to meet specific accuracy criteria of interest to the user; that is, criteria which can be used to determine whether a numerical simulation actually reflects physical reality. The goals of this project are to develop such criteria for making the simulations efficient, accurate and physically realistic, and to train future researchers in the underlying mathematics, computational science and multidisciplinary aspects of the applications.
