Boundary Element Calculations of Diffusion Equation

The boundary element method has become popular for solving elliptic equations and time‐dependent problems where time appears only in the boundary conditions. Its most often‐cited attribute is efficiency, although user convenience, its ability to solve singular problems, and the ease of solution in infinite regions probably outweigh the efficiency aspects. The boundary element method has been used in parabolic problems, but its advantages in that case are not as apparent. In this paper, three closely related techniques for solving the diffusion equation are explored. Although the method does not retain the feature of a strict boundary technique, since domain integrations are required, the user interface can retain the advantages of a boundary method. Because of the need for repeated integrations, or the need to integrate transcendental functions over a domain, depending on the solution method, the efficiency advantage of boundary elements is lost, at least in simple problems without singularities or infinite regions. A comparison with the finite element method illustrates that fact. Nevertheless, the boundary element remains a viable option for the solution of parabolic equations.