Domain Decomposition Methods for Coupled Models of Non-Newtonian Fluids and Solid Structures

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There has been increasing interest in numerical studies on fluids interacting with poroelastic structures; however, most numerical and analytical studies in this area have been focused on Newtonian fluids coupled with solid structures, and few analytical and numerical studies have been undertaken on the interaction of non-Newtonian fluids and elastic/poroelastic solids. Non-Newtonian fluid-poroelastic structure interaction (FPSI) problems are themselves of interest to related scientific and engineering communities, as the study of these strongly coupled nonlinear problems leads to a better understanding and a unifying description of complex real-life processes. This research will provide an underlying mathematical foundation for non-Newtonian flows in a multi-physical setting while promoting teaching, training graduate students, and involving undergraduates in research experiences. The proposed research will also benefit the biomedical, polymer, material, and oil industries by providing improved algorithms for the numerical simulation of important processes. The research activity lies in the development of rigorous, efficient, and stable numerical schemes for FPSI problems. Most current research results of FPSI problems discuss numerical methods with matching grids and time steps between subproblems in either a monolithic or partitioned setting. The goal of the research is to design domain decomposition methods for non-Newtonian FPSI systems that allow local space and time discretization. At the same time, the theoretical accuracy and stability properties of the numerical schemes will be analyzed. Specifically, we have five main objectives for this project: (i) develop and analyze parallel, accurate, efficient decoupling algorithms based on global-in-time, non-overlapping domain decomposition for non-Newtonian FPSI systems; (ii) develop and analyze accurate discretization and projection methods in time to effectively handle non-Newtonian FPSI systems with a large difference in local time scales; (iii) study the well-posedness of the viscoelastic FPSI system; (iv) develop and analyze accurate and efficient discretization methods for viscoelastic FPSI systems based on weighted least-squares finite element methods; (v) educate and train graduate and undergraduate students.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.