The geometrical basis of numerical stability


Douglas Arnold

IMA, University of Minnesota

Harvard University

Thursday, December 6, 2007


Talk at 4:30 p.m. in Science Center A (NOTE UNUSUAL PLACE)

Tea at 4:00 p.m. in the Math Lounge


Abstract:   The accuracy of a numerical solution to a partial differential equation depends on the consistency and stability of the discretization method. While consistency is usually elementary to establish, stability of numerical methods can be subtle, and for some key PDE problems the development of stable methods is extremely challenging. After illustrating the situation through simple but surprising examples, we will describe a powerful new approach--the finite element exterior calculus---to the design and understanding of discretizations for a variety of elliptic PDE problems. This approach achieves stability by developing discretizations which are compatible with the geometrical and topological structures, such as de Rham cohomology and Hodge decompositions, which underlie well-posedness of the PDE problem being solved.


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Posted: November 20, 2007    URL: