Higher-order boundary element methods for transient diffusion problems. Part I: Bounded flux formulation

Despite the significant number of publications on boundary element methods (BEM) for time-dependent problems of heat diffusion, there still remain issues that need to be addressed, most importantly accuracy of the numerical modelling. Although very precise for steady-state problems, the common boundary element methods applied to transient problems do not yield highly accurate numerical solutions. This paper investigates the reasons that prohibit achievement of a high level of accuracy for transient heat diffusion problems with continuous temperature and bounded heat flux solutions. In order to greatly enhance the commonly used boundary element formulations, we propose higher-order time interpolation functions, including quadratic and quartic approximations. We show that the use of higher-order time functions greatly reduces the numerical error concentrated in the corner regions, and results in very good uniformity of the flux and temperature distributions along the boundaries for problems where uniform distributions are expected. In order to highlight the importance of proper resolution both in time and space for the transient problems, we consider one- and two-dimensional formulations in this paper. High-order boundary elements using quartic shape functions, as well as high-order bi-quartic volume cells, are used to attain mesh-independent numerical solutions. We consider four transient heat diffusion, problems that possess exact solutions to investigate the convergence rate and accuracy of the higher-order boundary element formulations. A very high level of accuracy is possible for both one- and two-dimensional formulations. Additionally, we show that the accuracy of a commercially available finite-element code is far less than that of the boundary element methods for a given spatial and temporal discretization.