Mixed Finite Element Methods on Distorted Rectangular Grids

A new mixed nite element method on totally distorted rectangular meshes is introduced with optimal error estimates for both pressure and velocity This new mixed discretization ts the geometric shapes of the discontinuity of the rough coe cients and domain boundaries well This new mixed method also enables us to derive the optimal error estimates and existence and uniqueness of Thomas s mixed nite elementsmethod on distorted rectangular grids The lowest order Raviart Thomas mixed nite rectangular element method becomes a special case of both methods when all the elements are degenerated to parallelograms Introduction Let be a bounded convex polygonal domain in R with the boundary and We consider the homogeneous Dirichlet Neumann boundary elliptic problem r Krp f in p on Krp n on where K is a symmetric positive de nite matrix which is uniformly bounded below and above in n is the outward unit normal of Extension to inhomogeneous boundary condition problems is straight forward If we introduce a dependent vector valued variable u Krp then is equivalent to the following rst order partial di erential equation system K u rp in a r u f in b p on c u n on d In the simulation of uid ow in porous media such as groundwater contamination and petroleum reservoir simulation arises frequently in a system of partial di erential equations Here p stands for the hydraulic pressure or pressure and u stands for the uid velocity or Darcy veloc ity a represents Darcy s law and b is the mass conservation law which is one of the This research was supported in part by the Department of Energy under Contract No DE FG ER through the Institute for Scienti c Computation Texas A M University