Computational Mathematics Seminar

Wednesday, October 12, 3-4 pm, Thackeray 704

Speaker: Ivan Yotov

Title: Cell-centered polyhedral discretizations for flow

Abstract:
Mixed finite element (MFE) methods, mimetic finite difference (MFD)
methods, and multipoint flux approximation (MPFA) methods are three
popular classes of locally mass conservative methods that perform well
for flow problems with rough grids and coefficients.  While the MFE
and the MFD method require the solution of a saddle-point problem that
couples the scalar variable (pressure) and its flux (velocity), the
MPFA method only requires solving a cell-centered pressure system.
However, the non-variational formulation of the MPFA method presents
difficulties in its theoretical analysis.

We present a MFE method for elliptic problems that reduces to
cell-centered finite differences on simplicial and quadrilateral grids
and performs well for discontinuous full tensor coefficients.
Motivated by the MPFA method, where sub-edge fluxes are introduced, we
consider the lowest order Brezzi-Douglas-Marini (BDM) MFE method. A
special quadrature rule is employed that allows for local velocity
elimination and leads to a symmetric and positive definite
cell-centered system for the pressures.  Theoretical and numerical
results indicate second-order convergence for pressures at the cell
centers and first-order convergence for sub-edge fluxes on irregular
grids. Second-order convergence for edge fluxes is also observed
experimentally if the grids are sufficiently regular.  We also discuss
extensions of these results to hexahedral grids through the use of
enhanced BDM spaces. Finally, we present a related local-flux mimetic
finite difference method on polyhedral grids and discuss its
convergence properties on simplicial grids.