12 p.m. - 1 p.m. Location: FO 1.202
Fast Galerkin Methods for Parabolic Boundary Integral Equations
The boundary element method is a widely used technique for solving problems governed by elliptic PDEs. On the other hand, the application of integral equation methods to the heat equation or transient Stokes flow is much less developed and a topic of current research. Time dependence is reflected in the fact that boundary integral operators involve integrals over time in addition to integrals over the boundary surface. For the numerical solution this means that a time step involves a summation over space and the complete time history. Thus the naive approach has order N^2 M^2 complexity, where N is the number of unknowns in the spatial discretization and M is the number of time steps. We discuss a space-time version of the fast multipole method which reduces the complextiy to nearly NM.
A critical aspect of the success of boundary element methods is the choice of a proper discretization method. Since the thermal single layer operators is elliptic, the Galerkin method is unconditionally stable and optimally convergent. Each time step involves the solution of a linear system whose condition number is bounded with appropriate mesh refinement.
We also discuss the application of the methodology to industrial applications including modeling heat flows in hot forming tools and computing drag forces in microelectromechanical systems (MEMS).
John Zweck, 972-883-6699
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