A unified stabilized method for Stokes' and Darcy's equations

By Erik Burman, Peter Hansbo. 2007

216 citations recorded by the source corpus.

We use the lowest possible approximation order, piecewise linear, continuous velocities and piecewise constant pressures to compute solutions to Stokes equation and Darcy's equation, applying an edge stabilization term to avoid locking. We prove that the formulation satisfies the discrete inf-sup condition, we prove an optimal a priori error estimate for both problems. The formulation is then extended to the coupled case using a Nitsche-type weak formulation allowing for different meshes in the two subdomains. Finally, we present some numerical examples verifying the theoretical predictions and showing the flexibility of the coupled approach.

Subfields: Computational Mechanics, Mechanics of Materials

View Paper: https://www.semanticscholar.org/paper/150f318852e69acdb557ad9f17f7d7175c4871bd

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In Computational Mechanics

In Mechanics of Materials

Citations in the corpus

Convergence of a family of Galerkin discretizations for the Stokes–Darcy coupled problem (2011, background): “On the other hand, a priori error estimates for the matching case are developed in [8], whereas discontinuous Galerkin schemes for the nonmatching case are considered in [12] and [9].”
Navier-Stokes/Darcy Coupling: Modeling, Analysis, and Numerical Approximation (2009, methodology): “Under these conditions, the analysis of the coupled Stokes/Darcy problem has been studied in [40, 49, 53, 55, 56, 68, 69, 70, 71, 72, 88, 112, 113] in the steady case, and in [43, 124] in the time-dependent case.”
Mortar finite element discretization of a model coupling Darcy and Stokes equations (2008, background): “Note that a lot of recent works deal with the coupling of Darcy and Stokes equations in different frameworks, hence with other types of matching conditions such as the Beaver-Joseph-Saffman conditions, see [10,16,21,23,28,35] and the references therein.”
Time-dependent coupling of Navier–Stokes and Darcy flows (2013, methodology): “For instance, the reader can refer to the works of Arbogast and Lehr in [3], of Arbogast and Brunson in [2], of Burman and Hansbo in [7], of Cao et al.”
Stabilization of explicit coupling in fluid-structure interaction involving fluid incompressibility (2009, unknown): “By taking (vh, qh,wh, ẇh) = (uh, ph, ∂tηh, ∂tη̇h) , in (4) we have , d dt ( ρf 2 ‖uh‖20,Ωf + ρs 2 ‖η̇h‖20,Ωs + 1 2 as(ηh,ηh) ) + 2µ‖ (uh)‖20,Ωf + γµ h ‖uh − ∂tηh‖20,Σ − 2(1 + α)µ ∫ Σ (uh)n · (uh − ∂tηh) = 0.”
A mixed finite element method for Darcy flow in fractured porous media with non-matching grids (2012, methodology): “We also point out that different formulations using stabilized (Pk+1, Pk) finite elements (with interior penalties) for the Darcy’s equation have been studied in [9, 10], which allow a unified formulation in the framework of coupling with viscous flows.”
Topology optimization of creeping fluid flows using a Darcy–Stokes finite element (2006, methodology): “This leads to challenges in determining approximating function spaces that yield unified numerical stability of both flow models [20, 21 ].”
NON-MATCHING MORTAR DISCRETIZATION ANALYSIS FOR THE COUPLING STOKES-DARCY EQUATIONS (2007, background): “A similar example is presented in [8], where the term ÜNKÉ is replaced by ]Ë in the Stokes equations.”
A family of nonconforming elements for the Brinkman problem (2012, unknown): “These include, but are not limited to, stabilization methods [12, 11, 5] and augmented Lagrangian methods [9, 18].”
UNIFORMLY-STABLE FINITE ELEMENT METHODS FOR DARCY-STOKES-BRINKMAN MODELS (2008, background): “Roughly speaking, this assumption says discretely divergence-free velocity implies the almost everywhere divergence-free one.”