This proposal is for development of an effective hp-adaptive First-Order System Least-Squares (FOSLS) method for nonlinear partial differential equations (PDEs) that may exhibit singularities and other non-smoothness properties. The general FOSLS methodology is already a powerful tool for the solution of PDE-based problems in science and engineering. One of its main advantages is that it can be used to transform a given set of equations into a loosely coupled system of scalar equations that can be treated easily by multilevel finite elements. The reformulation is done by recasting the equations as a least-squares principle associated with a carefully derived first-order system. Another advantage of FOSLS is that the associated functional itself provides a natural sharp local error estimator, which can be employed for effective adaptive refinement. In previous work, the FOSLS methodology was developed and analyzed for problems that include linear elasticity, linear transport, and incompressible fluid dynamics. Under standard H2-type smoothness assumptions on the original PDE, FOSLS has been proved theoretically and numerically to exhibit optimal finite element and multigrid convergence properties. However, two principal difficulties currently prevent FOSLS from being applied effectively to more complicated problems: nonsmooth solutions and nonlinearity. The project aim is to develop an adaptive weighted-functional coupled with adaptive nested iteration schemes to treat these difficulties. The goal is to obtain a scheme that can solve complex nonlinear PDE systems with a total cost equivalent to a small number of relaxation steps on the finest grid.
Successful completion of this project will enhance our ability to model complex physical processes on high-end computing platforms. In particular, this project addresses computational models of coupled fluid/structure interactions that occur, for example, in biomechanical systems such as blood flow in compliant vessels. Other areas that could potentially benefit from this project are aerodynamics, astrophysics, geophysics and meteorology. As modern computer architectures become more complex, harnessing 10s of thousands of processors working together on a single, complex problem, it becomes even more important to develop optimal numerical algorithms. Successful completion of this project will widen the class of problems for which optimal algorithms exist.
