A posteriori error estimates for the Stokes equations: a comparison
Computer Methods in Applied Mechanics and Engineering
A posteriori error estimates for the Stokes problem
SIAM Journal on Numerical Analysis
Mixed and hybrid finite element methods
Mixed and hybrid finite element methods
A posteriori error estimation and adaptive mesh-refinement techniques
ICCAM'92 Proceedings of the fifth international conference on Computational and applied mathematics
A Posteriori Error Estimators for the Raviart--Thomas Element
SIAM Journal on Numerical Analysis
A posteriori error estimate for the mixed finite element method
Mathematics of Computation
A Posteriori Error Estimators for the Stokes and Oseen Equations
SIAM Journal on Numerical Analysis
An a posteriori error estimate for a first-kind integral equation
Mathematics of Computation
Coupling Fluid Flow with Porous Media Flow
SIAM Journal on Numerical Analysis
Mathematical and numerical models for coupling surface and groundwater flows
Applied Numerical Mathematics
A posteriori error estimators for mixed finite element methods in linear elasticity
Numerische Mathematik
Locally Conservative Coupling of Stokes and Darcy Flows
SIAM Journal on Numerical Analysis
Two-Sided A Posteriori Error Estimates for Mixed Formulations of Elliptic Problems
SIAM Journal on Numerical Analysis
Coupled Generalized Nonlinear Stokes Flow with Flow through a Porous Medium
SIAM Journal on Numerical Analysis
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In this paper we develop an a posteriori error analysis of a new conforming mixed finite element method for the coupling of fluid flow with porous media flow. The flows are governed by the Stokes and Darcy equations, respectively, and the transmission conditions are given by mass conservation, balance of normal forces, and the Beavers-Joseph-Saffman law. The finite element subspaces consider Bernardi-Raugel and Raviart-Thomas elements for the velocities, piecewise constants for the pressures, and continuous piecewise linear elements for a Lagrange multiplier defined on the interface. We derive a reliable and efficient residual-based a posteriori error estimator for this coupled problem. The proof of reliability makes use of suitable auxiliary problems, diverse continuous inf-sup conditions satisfied by the bilinear forms involved, and local approximation properties of the Clément interpolant and Raviart-Thomas operator. On the other hand, Helmholtz decomposition, inverse inequalities, and the localization technique based on triangle-bubble and edge-bubble functions are the main tools for proving the efficiency of the estimator. Up to minor modifications, our analysis can be extended to other finite element subspaces yielding a stable Galerkin scheme.