Fast integral equation methods for the modified Helmholtz equation

Abstract

Many problems in physics and engineering require the solution of the forced heat equation, $u_{t}-Delta u = F(xx,u,t)$, with Dirichlet boundary conditions. In this thesis, we solve such PDEs in two-dimensional, multiply-connected domains, with a twice continuously differentiable boundary. We discretize the partial differential equation (PDE) in time, known as Rothe\u27s method, leading to the modified Helmholtz equation, $u(xx)-alpha^{2}Delta u(xx)=g(xx,t)$. At each time step, solutions are written as the sum of a volume potential and a solution of $u(xx)-alpha^{2} Delta u(xx)=0$ with appropriate boundary conditions. The solution of the homogeneous PDE is written as a double layer potential with unknown density function. The density function satisfies a Fredholm integral equation of the second kind. Some advantages of integral equation methods are: the unknown function is defined only on the boundary of the domain, complex physical boundaries are easy to incorporate, the ill-conditioning associated with discretizing the governing equations is avoided, high-order accuracy is easy to attain, and far-field boundary conditions are handled naturally. The integral equation is discretized at $N$ points with a high-order, hybrid Gauss-trapezoidal rule resulting in a dense $N times N$ linear system. The linear system is solved using the generalized minimal residual method (GMRES). If the required matrix-vector multiplication is done directly, $mathcal{O}(N^{2})$ operations are required. This is reduced to $mathcal{O}(N)$ or $mathcal{O}(N log N)$ using the fast multipole method (FMM). To demonstrate the versatility of integral equation methods, the homogeneous problem is solved in bounded and unbounded, as well as simply- and multiply-connected domains. The volume integral is computed using a previously developed fast multipole-accelerated fourth-order method. This work is extended to general bounded domains and is coupled with the double layer potential. Applying Rothe\u27s method coupled with integral equation methods is tested on a collection of forced heat equation problems. This includes the homogeneous heat equation, a forced linear heat equation, and the Allen-Cahn equation