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Numerical approximation of an elliptic-parabolic equation arising in environment. (English) Zbl 1060.76075

Summary. We prove the convergence of a finite volume scheme for the Richards equation \(\beta(p)_t-\text{div}(\lambda(\beta(p)) (\nu p-\rho g) =0\), together with a Dirichlet boundary condition and an initial condition, in a bounded domain. We consider the hydraulic charge \(u=\frac{p}{\rho g}-z\) as the main unknown function, so that no upwinding is necessary. The convergence proof is based on the strong convergence in \(L^2\) of the water saturation \(\beta(p)\), which one obtains by estimating differences of space and time translates and applying Kolmogorov’s theorem.

MSC:

76M12 Finite volume methods applied to problems in fluid mechanics
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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