CAM Colloquium-- Johnny Guzman, Professor of Applied Mathematics, Brown University

Description

Title: Finite Element Exterior Calculus  (FEEC) with smoother spaces

Abstract: FEEC is now a well developed field of numerical analysis. It borrows from and makes connections with several other areas in mathematics (e.g.  algebraic topology, geometric integration). A central topic in FEEC are Whitney forms which were originally developed by H. Whitney to prove de Rham's Theorem. In fact, Whitney forms were later, independently,  discovered in two and three dimensions by J.-C. Nedelec and since then have been used in a wide variety of applications: electro-magnetism, solid mechanics, etc.   Whitney forms form a discrete de Rham complex of a simplicial decomposition. The regularity of this complex is in some sense minimal, whereas in some applications forms with more regularity are more natural. In recent years with collaborators, borrowing ideas from the spline community we have developed forms that are smoother that also fit into discrete de Rham complexes.  This led us to tackle two questions that were unresolved in our community: 1) Develop a discrete elasticity sequence on a simplicial triangulation in three dimensions and 2) justify why Lagrange elements on special meshes work for Maxwell eigenvalue problems.

Bio: Ph.D., 2005, Cornell University, Applied Mathematics

My main research interest lies in the area of numerical approximations to partial differential equations (PDEs). I work in devising new numerical methods for various PDEs and analyzing new or existing numerical methods.

Awards:

• Comfort and Urry Family Fund Prize, 2013
• NSF Postdoctoral Fellowship, 2005
• Ford Foundation and Cornell-Sloan Fellowships, 1999
• CSULB’s Outstanding Graduate in Mathematics, 1999