A Newton-Galerkin Algorithm for Variational Investigations. Focus: Nonlinear Elliptic BVP

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ABSTRACT:The proposed research primarily concerns finding, describing, and approximating solutions to semilinear elliptic zero-Dirichlet boundary value problems. While the Principal Investigator's first priority has been to prove existence and nodal structure theorems, his current efforts center on computational investigations. The PI will advance the numerical techniques used in the study of nonlinear differential equations in general and elliptic boundary value problems in particular. The PI's experiments support conjectures and reveal underlying structure as well as provide approximations. Secondary objectives include involving his students in the implementation of software solutions and refinement of algorithms. Lastly, the PI will prove convergence results relating to steepest descent, Newton's method, and Galerkin-type approximations. That these results may lead to analytical existence and nodal structure theorems is an additional source of motivation to the PI. The techniques to be used rely on the variational method, whereby solutions to the PDE are characterized as critical points of an associated nonlinear functional. The PI is currently in the process of decomposing function space into Newton flow-invariant manifolds and basins of attraction. These subsets of function space will be important to both numerical and analytical investigations.The proposed research is relevant to important areas of mathematics and science. Almost any scientific area concerns rates of change, hence differential equations. Many of these areas rely on the class of differential equations known as elliptic, and most of the difficult and physically significant problems are nonlinear. The PI's work is on the boundary of the computational and the analytical. An exciting new use of computational mathematics is in the investigation of nonlinear functional analysis. The PI has new techniques in nonlinear functional analysis for studying the underlying structure of this class of problem and believes that the techniques may generalize to an even wider class. Thus, funding this line of inquiry will benefit a large area of mathematics, soften or solve some long standing open problems in nonlinear elliptic differential equations, and be of use to many physical scientists.