Preconditioned space-time boundary element methods for the one-dimensional heat equation
For researchers in numerical methods for PDEs, this provides a novel preconditioning strategy for space-time boundary element methods, though currently limited to 1D.
The paper presents a preconditioned space-time boundary element method for the 1D heat equation, achieving efficient and robust preconditioning via boundary integral operators of opposite orders, with stability analysis that extends to higher dimensions.
In this note we describe a space-time boundary element discretization of the heat equation and an efficient and robust preconditioning strategy which is based on the use of boundary integral operators of opposite orders, but which requires a suitable stability condition for the boundary element spaces used for the discretization. We demonstrate the method for the simple spatially one-dimensional case. However, the presented results, particularly the stability analysis of the boundary element spaces, can be used to extend the method to the two- and three-dimensional problem.