Discontinuous Galerkin Methods for Problems with Fractional Derivatives

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In this effort the investigator and his student will consider developments of discontinuous Galerkin methods suitable for solving fractional differential equations. Motivated by a series of preliminary but very encouraging computational experiments, local discontinuous Galerkin methods for both fractional ordinary and partial differential equations will be considered. These initial computational observations supports a number of conjectures and these will guide the development and analysis of the proposed methods. While specific target applications are not considered detail, problems of a more practical character such as computational efficiency, multi-dimensional problems, and alternative formulations will be addressed in the latter part of the effort.While the notion of fractional calculus is as old as that of classical calculus introduced by Newton, the development and analysis of fractional calculus and fractional equations is not nearly as mature. However, during the last few decades fractional calculus has emerged as a natural and important description for a broad range of non-classical phenomena in the applied sciences and engineering. Examples can be found anomalous transport processes, sub-diffusion, and problems dominated by memory effects. Applications of such models are found in wide range of areas such as porous, visco-elastic, or biological flows, flow of crude oil in reservoirs, fusion plasma problems, modeling of properties of complex materials, financial markets etc. The planned activities seek to develop efficient and accurate computational techniques to allow application scientists and engineers to more effectively solve this important class of models.