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Finite volume element approximations of nonlocal reactive flows in porous media. (English) Zbl 0961.76050
Summary: We study finite volume element approximations for two-dimensional parabolic integro-differential equations arising in the modeling of nonlocal reactive flows in porous media. These types of flows are also called non-Fickian flows, and they exhibit mixing length growth. For simplicity, we consider only linear finite volume element metbods, although higher-order volume elements can be considered as well in this framework. It is proved that the finite volume element approximations are convergent with optimal order in $H^1$- and $L^2$-norm, and are superconvergent in a discrete $H^1$-norm. By examining the relationship between finite volume element and finite element approximations, we prove convergence in $L^\infty$- and $W^{1,\infty}$-norms. These results are also new for finite volume element methods for elliptic and parabolic equations.

76M12Finite volume methods (fluid mechanics)
76M10Finite element methods (fluid mechanics)
76S05Flows in porous media; filtration; seepage
76V05Interacting phases (fluid mechanics)
65M12Stability and convergence of numerical methods (IVP of PDE)
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