Singularities in the finite element approximation of two-dimensional diffusion problems

The solution of a two-dimensional elliptic boundary value problem with piecewise smooth external boundaries, interfaces, and diffusion coefficients typical of nuclear reactor structures is known to contain a singular part. The presence of singular functions in the neighborhood of each angular point for a given geometric configuration has important consequences on the convergence orders for approximate solutions of the problem. These consequences are analyzed both theoretically and numerically, in the framework of the finite element method. Some means are described to overcome the damaging effects of the singular points. A thorough numerical study of various reactor configurations extending from liquid-metal fast breeder reactors to pressurized water reactors shows that in the latter case, the use of high-order polynomials is partially unjustified, given the severe limitations on the convergence orders