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The main goal of this paper is to establish the convergence of mimetic discretizations of the first-order system that describes linear diffusion. Specifically, mimetic discretiza-tions based on the support-operators methodology (SO) have been applied successfully in a number of application areas, including diffusion and electromagnetics. These(More)
Flow simulations in porous media involve a wide range of strongly coupled scales. The length scale of short and narrow channels is on the order of millimeters, while the size of a simulation domain may be several kilometers (the richest oil reservoir in Saudi Ara-bia, Ghawar, is 280 km ×30 km). The per-meability of rock formations is highly heterogeneous(More)
Class of problems We consider the 2D steady-state linear diffusion ∇ · J = Q(x, y) J = −D(x, y)∇φ subject to the Dirichlet boundary condition φ(x, y) = g(x, y). Key issues influencing the discretization and the linear solver include: ­ D(x, y) is defined on, and possibly discontinuous on, a very fine scale ­ coarse-scale view of the fine-scale structure may(More)
SUMMARY Although there have been significant advances in robust algebraic multigrid methods in recent years, numerical studies and emerging hardware architectures continue to favor structured-grid approaches. Specifically, implementations of logically structured robust variational multigrid algorithms, such as the Black Box Multigrid (BoxMG) solver, have(More)
– The Laplace-Beltrami system of nonlinear, elliptic, partial differential equations has utility in the generation of computational grids on complex and highly curved geometry. Discretization of this system using the finite element method accommodates unstructured grids, but generates a large, sparse, ill-conditioned system of nonlinear discrete equations.(More)
Certain classes of nodal methods and mixed-hybrid nite element methods lead to equivalent , robust and accurate discretizations of 2 nd order elliptic PDEs. However, widespread popularity of these discretizations has been hindered by the awkward linear systems which result. The present work overcomes this awkwardness and develops preconditioners which yield(More)