Relationships Among Some Locally Conservative Discretization Methods Which Handle Discontinuous Anisotropic Coefficients on Deformed Grids

Thursday, May 13, 2004 - 11:00am - 11:50am
Keller 3-180
Thomas Russell (National Science Foundation)
This talk presents the relationships between some numerical methods suitable for a heterogeneous elliptic equation of the form - div (K(x) grad p) = q, motivated by applications to subsurface flow (pressure or potential) equations. The methods discussed are the classical Raviart-Thomas mixed finite element method (MFEM), the control-volume mixed finite element method (CVMFEM), the support operators method (SOM), the enhanced cell-centered finite difference method (ECCFDM), and the multi-point flux-approximation (MPFA) control volume method. These methods are all locally mass conservative, and handle general irregular grids with anisotropic and heterogeneous discontinuous conductivity K(x). In addition to this, the methods have in common a weak continuity in the pressure across the edges, which in some cases corresponds to Lagrange multipliers. This weak continuity appears to be an essential property for the accuracy of these methods.

While the methods are applicable in two and three dimensions, the details of the above relationships are presented for logically rectangular quadrilateral grids in 2D. Issues of deformed grids are substantially more complex in 3D. An example (Naff, Russell, and Wilson, Computational Geosciences, 2002) is presented in which the Piola-transformed lowest-order Raviart-Thomas spaces fail to contain the constant-velocity vector fields for a hexahedron that is a trilinear image of a reference cube.

This represents joint work with R.A. Klausen and R. Winther of the University of Oslo, Norway.