Abstract
In this paper, we develop an a posteriori error analysis for the stationary Stokes-Darcy coupled problem approximated by conforming the finite element method on isotropic meshes inℝd,d∈2,3. The approach utilizes a new robust stabilized fully mixed discretization. The a posteriori error estimate is based on a suitable evaluation on the residual of the finite element solution plus the stabilization terms. It is proven that the a posteriori error estimate provided in this paper is both reliable and efficient.
Highlights
There are many serious problems currently facing the world in which the coupling between groundwater and surface water is important
One important issue in the modeling of the coupled Darcy-Stokes flow is the treatment of the interface condition, where the Stokes fluid meets the porous medium
To capture the fully mixed technique in the porous medium region linear Lagrangian elements, P1 has been used for hydraulic head and Brezzi-Douglas-Marini (BDM1) piecewise constant finite elements have been used for Darcy velocity
Summary
There are many serious problems currently facing the world in which the coupling between groundwater and surface water is important. The works on the strongly coupled formulation have been based on the development of a unified discretization, namely, the Stokes side and the Darcy side are discretized using the same finite element. To our best knowledge, they did not talk about the adaptative method for the fully mixed discretization proposed in [16] In this case, our main objective is to perform an a posteriori error analysis by constructing reliable and efficient indicator errors. The difference between our paper and the reference [40] is that our discretization uses MINI elements (P1b − P1) to approximate the velocity and pressure in the conduit for Stokes equations; P1 -Lagrange elements to approximate hydraulic (piezometric) head and Brezzi-Douglas-Marini (BDM1) piecewise constant finite elements have been used for Darcy velocity.
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