The goal of this work is to develop a fast method for solving Galerkin discretizations of boundary integral formulations of the heat equation. The main contribution of this work is to devise a new fast algorithm for evaluating the dense matrices of the discretized integral equations.
Similar to the parabolic FMM, this method is based on a subdivision of the matrices into an appropriate hierarchical block structure. However, instead of an expansion of the kernel in both space and time we interpolate kernel in the temporal variables and use of the adaptive cross approximation (ACA) in the spatial variables.
The second objective of this dissertation is to extend the software package BEM++, which was written for elliptic operators, to handle thermal layer potentials. To that end, we use the package's ACA implementation for the space variables and develop a python interface to handle time dependence.
The validity of our implementation is tested for several problems with known solution and for problems with geometries that are more close to realistic engineering applications. The results demonstrate that the fast method can reproduce the theoretical convergence rates of the direct method while improving the computational cost to nearly linear complexity in the number of discretization parameters.
Creative Commons License
This work is licensed under a Creative Commons Attribution-Noncommercial 4.0 License
Wang, Sihao, "The Boundary Element Method for Parabolic Equation and Its Implementation in BEM++" (2020). Mathematics Theses and Dissertations. 7.