There are many practical physical situations that can be modeled by partial differential equations in which some of the coefficients in the problem, e.g. those describing material properties, are discontinuous across an interface. Such problems arise in a very broad range of scientific and engineering disciplines, including computational biology, ground water flow and reservoir simulation, environmental remediation studies, crystal growth, wave propagation, sedimentation phenomena, and the preparation of nuclear fuel rods. Such problems cause notorious difficulties for classic numerical methods as a direct result of the lack of smoothness. At the same time, numerical methods for efficiently solving interface problems using a fixed Cartesian grid have attracted considerable attention because they offer a number of computational advantages and a number of approaches have been pursued. Unfortunately, regular-shape discretizations are generally problematic because the interface ?cuts? through the cells without respecting the regular geometry of the discretization, which has a strongly negative impact on the accuracy of the resulting approximations. Consequently, it is critically important to provide computational error estimates that quantify the accuracy of a computed quantity of interest in terms of various sources of discretization and modeling error.
This project will develop variational finite element frameworks for several discrete interface methods that identify both a discretization and a modeling component to the model and use the variational framework to derive accurate a posteriori error estimates for specified quantities of interest. Both stationary and evolutionary problems in two and three space dimensions will be considered. The development of efficient adaptive discretization methods for both stationary and evolution problems will be addressed. The methodology and analytic tools to be developed in this proposal will provide a powerful tool for the systematic treatment of problems in which interfaces have complex geometry and the material properties vary considerably on a scale smaller than the overall scale of the discretization. The project will yield a systematic approach to deriving computable and accurate error estimates for quantities of interest and further, provide detailed information about the relative contributions to the error arising from discretization and modeling.
Finite element methods are a well-established and popular technique for solving systems of partial differential equations which provide mathematical descriptions of physical phenomena, for example, the flow in a porous medium. Adjoint-based a posteriori analysis seeks to estimate the error in a quantity of interest based on the result of a specific numerical calculation. A posteriori methods were first developed for relatively simple physical systems, yet through computation we now seek to understand complex systems involving multiple physical processes which often evolve on distinctly different time scales. In many real-world situations, the model parameters and even the computational domains may not be known with complete precision and so the computational problem is only known with some degree of uncertainty. Complicating matters, particular numerical approaches, such as finite volume methods and explicit time integration techniques, have become entrenched within some scientific communities. If they are to become useful beyond a relatively narrow range of application areas, new a posteriori techniques must be developed in order to accommodate these different circumstances. This work has extended the mathematically appealing adjoint-based a posteriori ideas from finite-element methods for single physics problems to a range of computational problems encountered in practice. Further, the insights afforded in to the relative sizes of the contributions from various sources of error can be used to construct appropriate adaptive computational strategies. The major focus of this project has been to ground water flow (flow in a porous medium) in which the properties of the medium are known to change abruptly across an interface, yet the location of the interface is known through only a small number of imprecise measurements. These problems are traditionally solved using finite volume techniques. Our approach is to construct and analyse an equivalent finite element method. The error in a quantity of interest based on a specific numerical solution is then separated in to a component that arises due to the numerical technique and a component that arises due to the uncertainty in the interface location. A separation of the error in to these two different components is essential to determine whether the solution can be improved through a better numerical approximation technique or a through a better knowledge of the interface location. A similar approach, namely constructing and analysing an equivalent finite element (or variational) scheme, was used to conduct adjoint-based a posteriori error analyses for some common explicit integration schemes for ordinary differential equations. The schemes for which accurate a posteriori error estimators were developed included not only low-order schemes such as explicit Euler and trapezoidal rule, but the very popular multistep Adams-Bashforth fourth-order method and the multistage Runge-Kutta fourth-order method.
