On singularities in the solution of three‐dimensional Stokes flow and incompressible elasticity problems with corners

Abstract

AbstractIn this paper, a numerical procedure is presented for the computation of corner singularities in the solution of three‐dimensional Stokes flow and incompressible elasticity problems near corners of various shape. For obtaining the order and mode of singularity, a neighbourhood of the singular point is considered with only local boundary conditions. The weak formulation of this problem is approximated using a mixed u, p Galerkin–Petrov finite element method. Additionally, a separation of variables is used to reduce the dimension of the original problem. As a result, the quadratic eigenvalue problem (P+λQ+λ2R)d=0 is obtained, where the saddle‐point‐type matrices P, Q, R are defined explicitly. For a numerical solution of the algebraic eigenvalue problem an iterative technique based on the Arnoldi method in combination with an Uzawa‐like scheme is used. This technique needs only one direct matrix factorization as well as few matrix–vector products for finding all eigenvalues in the interval 𝓈(λ) ∈ (−0.5, 1.0), as well as the corresponding eigenvectors. Some benchmark tests show that this technique is robust and very accurate. Problems from practical importance are also analysed, for instance the surface‐breaking crack in an incompressible elastic material and the three‐dimensional viscous flow of a Newtonian fluid past a trihedral corner. Copyright © 2004 John Wiley & Sons, Ltd.