Fast integral equation methods for the modified Helmholtz equation

Talk Abstract We present an efficient integral equation method approach to solve the forced heat equation, ut(x) − ∆u(x) = F (x, u, t), in a two dimensional, multiply connected domain, with Dirichlet boundary conditions. We first discretize in time, which is known as Rothe’s method, resulting in a non-homogeneous modified Helmholtz equation that is solved at each time step. We formulate the solution to this equation as the sum of a volume potential and a double layer potential. Both potentials are solved using the Fast Multipole Method (FMM) resulting in a O(N) method where N is the total number of discretization points on the boundary and in the domain. We demonstrate our approach on the heat equation and the Allen-Cahn equation.