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Convergence of a mixed finite element-finite volume method for the two phase flow in porous media. (English) Zbl 0899.76261

Summary: As a model problem for the miscible and immiscible two phase flow, we consider the following system of differential equations: \(\text{div }u(x, t)= 0\), \(u(x, t)= -a(c,x)(\nabla p(x, t)+ \gamma(c, x))\), \(\partial_tc(x, t)+ \text{div}(u(x, t)c(x, t))- \text{div}(\varepsilon\nabla c(x, t))= f(x, t)\) with \((x, t)\in\Omega\times (0,T)\). Here \(u\) denotes the Darcy velocity, \(p\) the pressure and \(c\) the concentration of one phase of the fluid. Considering density driven flow or immiscible flow of water and oil in a reservoir, the convection of the concentration is dominant to the diffusion. Thus we have to look at this system of partial differential equations as at a singular perturbed problem in \(\varepsilon\). For small diffusions \((0<\varepsilon\ll 1)\) standard Galerkin finite element approximations do not produce stable solutions. Therefore we propose a combined mixed finite element – finite volume discretization, specifically to handle this convection dominated diffusion problem. Taking into account the dependence on the diffusion parameter \(\varepsilon\), we prove a convergence result for this first order scheme on triangular meshes.

MSC:

76M10 Finite element methods applied to problems in fluid mechanics
76M25 Other numerical methods (fluid mechanics) (MSC2010)
76T99 Multiphase and multicomponent flows
76S05 Flows in porous media; filtration; seepage
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
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