Fractional diffusion equations provide an adequate description of transport processes that exhibit anomalous diffusion, which cannot be modeled properly by classical second-order diffusion equations. However, fractional diffusion equations introduce severe computational, numerical, and mathematical difficulties which have not been encountered in the context of second-order equations: (i) Fractional diffusion equations lead to numerical methods with dense or full coefficient matrices, which makes realistic three-dimensional simulations computationally intractable! (ii) Fractional diffusion operators are non-local and the adjoint of a fractional differential operator is not the negative of itself, which significantly complicates the mathematical analysis. The objectives of this proposal are as follows: (i) Develop fast numerical methods for fractional diffusion equations with significantly improved computational efficiency and memory requirement while retaining the stability and accuracy of standard methods. (ii) Develop efficient preconditioners for the fast numerical methods, so that the convergence of the preconditioned linear system is independent of the mesh size. (iii) Conduct corresponding mathematical and numerical analysis for the proposed fast methods.
Diffusion processes are ubiquitous and occur in nature, sciences, social sciences, and engineering. Sample applications include how water and nutrients travel through membranes in living organisms, how mosquitoes spread malaria, how copiers and laser printers work, and how contaminants in groundwater are transported, as well as the signaling of biological cells, foraging behavior of animals, and finance. Fick first sat up the diffusion equation in 1855. But it was Einstein who derived the diffusion equation from first principle as part of his work on Brownian motion. In last few decades it was found that increasingly more diffusion processes cannot be properly modeled by classical diffusion equations. These discoveries have profound consequences. For example, recent modeling by fractional advection-diffusion equations indicate that remediation of contaminated aquifers may take decades or centuries longer than previously predicted by the classical advection-diffusion equations. Hence, further investigations are crucial. The results of this work will be applicable to a wide range of applications. The proposed research activities will also provide advanced interdisciplinary training to graduate and undergraduate students. All of these activities will have broad and long-lasting impacts and contribute directly to the intellectual infrastructure of the nation.
