2015-02-09

# Fast Galerkin Methods for Parabolic Boundary Integral Equations

## Prof. Johannes Tausch, Southern Methodist University Dallas, Texas (USA)

9 Mar 2015, 17:00–18:30; Location: S4|10-1

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 time dependent problems 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 complexity to nearly $$N M$$.

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 three-dimensional transient Stokes flow. Perhaps surprisingly, this is not straight forward, because of different properties of the fundamental solutions of the heat and Stokes equations.

Category: CE Seminar