On numerical solution stability of the Laplace equation in absence of the Dirichlet boundary condition: Benchmark problem proposal

Abstract

A fundamental 1D steady state diffusion problem using only third type boundary condition also named as the Robin or Fourier type is analysed. In the heat transfer problems this boundary condition is called convective boundary condition using heat transfer coefficient ( α ) as a constant. The main idea of the proposed Benchmark problem is to test the solution accuracy in both limiting cases of α approaching to zero and to infinity. The infinite case limits to Dirichlet boundary condition and should not be the problem, while the zero limit leads to trivial solution of zeros in the absence of the Dirichlet boundary condition. The application of the Benchmark would be in freeze drying, where similar boundary conditions occur. The presented BEM numerical solution failed for α > 1 0 13 in double precision computation for two 2D formulations: cartesian and axisymmetric. The low limit results failed for α < 1 0 − 13 for cartesian formulation. This is expected, while the failure of axisymmetric formulation results already at moderate low α < 1 0 − 3 value is not. The solution stability and accuracy are found strong dependent to the fundamental solution integrals accuracy. The effect of system matrix preconditioner is also studied.