A posteriori error analysis of a fully-mixed formulation for the Brinkman–Darcy problem

Abstract

We develop the a posteriori error analysis for a mixed finite element method applied to the coupling of Brinkman and Darcy equations in 3D, modelling the interaction of viscous and non-viscous flow effects across a given interface. The system is formulated in terms of velocity and pressure within the Darcy subdomain, together with vorticity, velocity and pressure of the fluid in the Brinkman region, and a Lagrange multiplier enforcing pressure continuity across the interface. The solvability of a fullymixed formulation along with a priori error bounds for a finite element method have been recently established in Álvarez et al. ( Comput Methods Appl Mech Eng 307:68-95, 2016). Here we derive a residual-based a posteriori error estimator for such a scheme, and prove its reliability exploiting a global inf-sup condition in combination with suitable Helmholtz decompositions, and interpolation properties of Clément and Raviart-Thomas operators. The estimator is also shown to be efficient, following a localisation strategy and appropriate inverse inequalities. We present numerical tests to confirm the features of the estimator and to illustrate the performance of the method in academic and application-oriented problems.

A posteriori error analysis of a semi‐augmented finite element method for double‐diffusive natural convection in porous media

“…and so, by their own definitions the norms || Flow || and || Trans ||, can be expressed as We start this section by recalling a few instrumentals results that will be employed to obtain a suitable upper bound for || Flow || by applying some techniques from previous works [1,3,5,21,25]. More precisely, we will make use of the classical properties of the usual Raviart-Thomas interpolator, the approximation properties of the Clément interpolation operator, and a stable Helmholtz decomposition of the space H 0 (div; Ω).…”

“…We start this section by recalling a few instrumentals results that will be employed to obtain a suitable upper bound for $$$ \left\Vert {\mathcal{R}}^{\mathtt{Flow}}\right\Vert $$$ by applying some techniques from previous works [1, 3, 5, 21, 25]. More precisely, we will make use of the classical properties of the usual Raviart‐Thomas interpolator, the approximation properties of the Clément interpolation operator, and a stable Helmholtz decomposition of the space $$$ {\mathbb{H}}_0\left(\mathbf{\operatorname{div}};\Omega \right) $$$.…”

A splitting method for the numerical simulation of free surface flows with sediment deposition and resuspension

“…Then Q T , the space-time domain containing the liquid is defined by 3 and the pressure field p ∶ Q T → R satisfy the time-dependent, incompressible Navier-Stokes equations, with variable density and viscosity coefficients, and modified by an additional Darcy-like reaction term. 35,36 Finally, the sediment concentration…”

“…Then $$$ {Q}_T $$$, the space‐time domain containing the liquid is defined by $$$ {Q}_T=\left\{\left(\mathbf{x},t\right)\in \Lambda \times \left(0,T\right):\varphi \left(\mathbf{x},t\right)=1\right\} $$$. The velocity field $$$ \mathbf{v}:{Q}_T\to {\mathbb{R}}^3 $$$ and the pressure field $$$ p:{Q}_T\to \mathbb{R} $$$ satisfy the time‐dependent, incompressible Navier–Stokes equations, with variable density and viscosity coefficients, and modified by an additional Darcy‐like reaction term 35,36 . Finally, the sediment concentration $$$ {f}_s:{Q}_T\to \left[0,{f}_{s_{CR}}\right] $$$ where $$$ {f}_{s_{CR}} $$$ is the maximal sediment solid fraction, satisfies a nonlinear conservation law.…”

Vorticity‐pressure formulations for the Brinkman‐Darcy coupled problem

“…If the permeability of the viscous domain goes to infinity, one readily recovers the classical Stokes flow, and the literature is populated with numerous formulations and methods to solve the Stokes‐Darcy and Navier‐Stokes–Darcy equations (e.g., and the references therein). In contrast, dedicated Brinkman‐Darcy models have been studied in Braack and Schieweck, Ervin and coworkers, Lesinigo and coworkers (using velocity‐pressure formulations), whereas the setting described above (including also the Brinkman vorticity) has been proposed only quite recently (along with a fully mixed finite element method solving for vorticity‐velocity‐pressure on the viscous domain and velocity‐pressure on the nonviscous domain and its a posteriori error analysis developed in Alvarez and coworkers ).…”