Title:

Sparse spacetime boundary element methods for the heat equation

The goal of this work is the efficient solution of the heat equation with Dirichlet or Neumann boundary conditions using the Boundary Elements Method (BEM). Efficiently solving the heat equation is useful, as it is a simple model problem for other types of parabolic problems. In complicated spatial domains as often found in engineering, BEM can be beneficial since only the boundary of the domain has to be discretised. This makes BEM easier than domain methods such as finite elements and finite differences, conventionally combined with timestepping schemes to solve this problem. The contribution of this work is to further decrease the complexity of solving the heat equation, leading both to speed gains (in CPU time) as well as requiring smaller amounts of memory to solve the same problem. To do this we will combine the complexity gains of boundary reduction by integral equation formulations with a discretisation using wavelet bases. This reduces the total work to O(hₓ(d1)), when the solution of the linear system is performed with linear complexity. We show that the discretisation with a wavelet basis leads to a numerically sparse matrix. Further, we show that this matrix can be compressed without losing accuracy of the underlying Galerkin scheme. This matrix compression reduces the number of nonzero matrix entries from O(N2) to O(N). Thus, we can indeed solve the linear system in linear time. It has been shown theoretically that using sparse grid methods leads to considerably higher convergence rates in the energy norm of the problem. In this work we will show that the convergence can be further improved for some choices of polynomial degrees by using more general sparse grid spaces. We also give numerical results to verify the theoretical bounds from [Chernov, Schwab, 2013].
