Collaborative Research: Adaptive Methods and Finite Element Exterior Calculus for Nonlinear Geometric PDE

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The primary technical aim of this project is to develop general approximation theory and reliable, convergent adaptive methods for the intrinsic discretization of a general class of nonlinear geometric elliptic and evolution PDE on Riemannian 2- and 3-manifolds. The investigators will exploit the variational crimes framework they have developed for the finite element exterior calculus (FEEC), extending the FEEC to nonlinear elliptic problems, to problems on hypersurfaces, and to nonlinear parabolic and hyperbolic problems. This framework will aid in the design, development, and convergence analysis of AFEM algorithms for use with FEEC. This approach will allow for a more natural and general treatment of geometric error due to variational crimes in a posteriori analysis, following their recent approach for a priori analysis. After obtaining a solid theoretical framework for a posteriori analysis, yielding a posteriori error estimates and local indicators, they will develop and analyze adaptive finite element methods (AFEM) within the extended FEEC framework. The convergence analysis approach will be based on their recent published work on AFEM convergence analysis for mixed formulations of linear elliptic problems. The overall goal is to develop a complete AFEM convergence theory in FEEC, complementing the recently developed contraction frameworks for non-mixed formulations of Poisson-type problems and semilinear generalizations. Both prototype and production implementations will be produced, using the opensource FETK ToolKit, and the resulting software will be used in ongoing collaborations with physical scientists and engineers.The investigators will study and develop methods for the approximate solution of systems of stationary and evolution partial differential equations (PDE) arising at the intersection of mathematical physics and geometric analysis. Such systems of equations, known as Geometric PDE, appear in a wide range of physical and mathematical problems; examples include Maxwell's equations (or more generally the Yang-Mills equations), Einstein's field equations, and other Hamiltonian systems. The Cauchy (or initial-value) formulation for such systems yields a constrained evolution system containing non-dynamical equations. These non-dynamical geometric PDE are of great interest in their own right; examples include the Yamabe problem, the Hamiltonian and momentum constraints in the Einstein equations, and the Monge-Ampere equations, among others. If our goals are achieved, the results of this project will have a broad impact on areas of mathematics such as geometric analysis, as well as in astrophysics and general relativity. The methods developed here will contribute to the advancement of numerical methods for complex three-dimensional constrained nonlinear dynamical simulations. The simulation technology we produce will provide powerful tools for the exploration of models in astrophysics and relativity as well as in some areas of pure mathematics such as geometric analysis. Graduate students involved in the project will be co-trained by both investigators; this will involve regular interaction between the members of the teams at both partner institutions. The PI has previously collaborated on such a shared training structure with great success on past projects; this shared training and transfer of knowledge and skills between the two research groups will be an invaluable research resource to both groups.