Coefficients of the singularities of elliptic and parabolic problems in domains with edges

The solution of an elliptic or a parabolic boundary value problem on a polyhedral cylinder (or a polyhedral domain) has a decomposition into a regular part and a singular one. Roughly speaking, this singular part is the superposition along the edge of a finite number of the same 2D-singular functions; so that for each 2D-singular function S, its multiplicative factor c is a function of the distance to the edge r but of the edge variable z too. The trace Phi(z) = c(0, z) on the edge of that function c is called the coefficient of the singularity S. For a large class of operators, we give different expressions for this coefficient Phi. Contrary to [19], our results hold without any restriction on the number of eigenvalues on a certain strip, they are also valid for the Lame system and for parabolic problems; moreover the exact form of the function c as the z-convolution of Phi with an explicit kernel K is preserved. The numerical approximation of that coefficients is also considered as in [4].