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2007 Harold J. Gay Lecture Series: "The Geometrical Basis of Numerical Stability" by Professor Douglas N. Arnold

Monday, February 26, 2007
4:00 pm

Location:

Professor Douglas N. Arnold

INSTITUTE FOR MATHEMATICS AND ITS APPLICATIONS, UNIVERSITY OF MINNESOTA

 

The Geometrical Basis of Numerical Stability

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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