For the numerical approximation of partial differential equations (PDEs), a balance is sought between the approximation properties (accuracy, consistency, stability, etc.), the solution time (solver speed, implementation efficiency), and robustness (scalability and applicability). To this end, the principle goal in this proposal is to develop a high-order discretization framework amenable to fast solution techniques in a multilevel setting. The application focus of the proposal is motivated by two core problems: neutrophil chemotaxis in the blood stream and cellular mechanics in microcirculation. Principally, these models are governed by coupled anisotropic diffusion-convection-reaction equations and coupled Stokes equations. The efficient and effective numerical solutions of these complex equations is central to the proposed work. This includes the development of an effective discontinuous least-squares spectral element method, a comparison with popular strategies such as discontinuous Galerkin, and the development of an integrated algebraic multigrid preconditioner for use with high-order spectral elements in this situation. Ultimately, this work establishes a theoretical and computation base for further research in discontinuous least-squares methods and high-order preconditioning. Moreover, an intrinsic element of this project is the integration of new methods in numerical PDEs and iterative methods into the existing scientific computing curriculum to help train future computational scientists.
As physical models grow in complexity and high-performance computing environments grow in speed, so do the demands on the underlying mathematical algorithms. The goal of this project is to make progress toward a more generalized mathematical framework that encompasses more layers of the entire simulation tool chain. Large-scale computational analysis is a critical experimental component in many areas of the physical sciences and yet, computational scientists are limited in their tool set. Olson proposes to develop a multilevel approximation method for core applications, such as cellular behavior in the blood stream, and to expand wider adaptation of new methods in the field through outreach and education. The proposed research methodology promotes conformity with the physics of the problem, allows for a natural and extensible computational implementation, and yields an accurate and efficient solution. This project will develop new steps for the multilevel methodology, disseminate the computational tools to the broader scientific and computing community, and train students and scientists on using these emerging computational technologies.
