hp-adaptive FOSLS Methods for Nonlinear PDE Problems with Singularities

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This proposal is for development of an effective hp-adaptive First-OrderSystem Least-Squares (FOSLS) method for nonlinear partial differentialequations (PDEs) that may exhibit singularities and other non-smoothness properties. The general FOSLS methodology is already a powerful tool forthe solution of PDE-based problems in science and engineering. One of itsmain advantages is that it can be used to transform a given set ofequations into a loosely coupled system of scalar equations that can betreated easily by multilevel finite elements. The reformulation is doneby recasting the equations as a least-squares principle associated with a carefully derived first-order system. Another advantage of FOSLS isthat the associated functional itself provides a natural sharp local errorestimator, which can be employed for effective adaptive refinement. Inprevious work, the FOSLS methodology was developed and analyzed forproblems that include linear elasticity, linear transport, andincompressible fluid dynamics. Under standard H2-type smoothnessassumptions on the original PDE, FOSLS has been proved theoretically and numerically to exhibit optimal finite element and multigridconvergence properties. However, two principal difficulties currentlyprevent FOSLS from being applied effectively to more complicated problems:nonsmooth solutions and nonlinearity. The project aim is to develop anadaptive weighted-functional coupled with adaptive nested iterationschemes to treat these difficulties. The goal is to obtain a scheme thatcan 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 modelcomplex physical processes on high-end computing platforms. In particular,this project addresses computational models of coupled fluid/structureinteractions that occur, for example, in biomechanical systems such asblood flow in compliant vessels. Other areas that could potentiallybenefit from this project are aerodynamics, astrophysics, geophysics andmeteorology. 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 optimalnumerical algorithms. Successful completion of this project will widen theclass of problems for which optimal algorithms exist.