Speaker: Marc Gerritsma (TU Delft)
Title: Mimetic Spectral Element Methods

Speaker: Benjamin Sanderse (CWI)
Title: CFD of wind farms

Date: 26 May 2009
Time: 10.30-12.30h, tea starting at 10h
Room: M279

Abstract:  
Mimetic schemes or compatible discretizations methods reconnect partial
differential equations with the geometric objects they are associated
with. It provides a unifying framework in which seemingly different
branches of numerical analysis turn out to have a common basis. In this
talk we will consider the Poisson equation -- or more generally steady
diffusion problems. If we write this as a first-order system and want to
apply the finite element method, we are confronted with the problem of
chosing appropriate discrete function spaces in order to satisfy the
inf-sup condition. An alternative approach is to use the support-operator
finite-difference method proposed by Hyman, Shashkov and Steinberg,
in which the gradient operator is implicitly defined as the transpose of
the divergence operator.

The mimetic approach allows one to find compatible discrete function
spaces for the mixed finite-element method purely based on geometrical
considerations. For low-order elements, we retrieve the well-known
Nedelec elements and Thomas-Raviart elements, but the geometric
interpretation allows one to extend these H(curl)- and
H(div)-conforming elements to higher order.

Compatible function spaces are obtained by employing two dual grids. On
the primal grid the divergence equation can be exactly satisfied and on
the secondary grid the gradient equation can be exactly satisfied. The
numerical solution of the Poisson equation is then reduced to connecting
the velocities on both grids. If we minimize the difference between
these two velocity fields in the L²-norm (least-squares minimization),
we obtain the support-operator method.

This paradigm also demonstrates that the two governing equations are
purely toplogical and thus metric free (i.e. independent of any choice
of basis functions). It is only when we establish the connection between
the dual grids that the metric becomes important.