Title: A new hp method for the -grad(div) and Stokes operators in non-Cartesian geometry

Speaker: Ralf Gruber, EPFL

Date/Time: Friday, March 3, 2006, 10:00 – 11:00 am

Location: Building 980, Room 95 (Sandia NM)

Brief Abstract: The grad(div) operator has an infinitely degenerate eigenvalue l=0 with eigensolutions satisfying exactly the constraints div(u)=0. In the case of a Cartesian mesh it is possible to find a perfect blend of polynomials in the different directions such that a primal (or energy) formulation of the operator leads to the degenerate eigenvalue and to constraints that are identically satisfied.  If the mesh is not further Cartesian, coordinate transformations have to be made to transform general quadrangles into a square. The div operator then includes all the partial derivatives, and no standard finite element or high-order approach can be found that can reproduce the degenerate eigensolution within the exact degeneracy. There are indeed non-physical eigensolutions with eigenvalues l>0 that pollute the discrete spectrum. A new non-conforming, non-polluting hp method can remedy this effect. In this method, all the different terms in a variational form are represented by the same polynomials and have the same regularities across element borders.  The same pollution-free approach is applied to solve the Stokes eigenvalue problem in two steps. First, the kernel of the -grad(div) operator is computed. These eigenmodes are used as basis functions to discretize the Laplacian operator. The obtained spectrum only includes the Stokes modes with the exact degeneracies. The eigenvalues converge exponentially in p and in O(h**(2p)) in h.  This new approach has also been successfully applied to the -curl(curl) and the MHD operators.

CSRI POC: Rolf Riesen, (505) 845-7363