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Download in PDF Format A posteriori error analysis for a sedimentation-consolidation system∗ Mario Alvarezy Gabriel N. Gaticaz Ricardo Ruiz-Baierx Abstract In this paper we develop the a posteriori error analysis of an augmented mixed{primal finite el- ement method for the 2D and 3D versions of a stationary flow and transport coupled system, typically encountered in sedimentation-consolidation processes. The governing equations consist in the Brinkman problem with concentration-dependent viscosity, written in terms of Cauchy pseudo- stresses and bulk velocity of the mixture; coupled with a nonlinear advection { nonlinear diffusion equation describing the transport of the solids volume fraction. The solvability of the continuous mixed{primal formulation along with a priori error estimates for a finite element scheme using Raviart-Thomas spaces of order k for the stress approximation, and continuous piecewise polyno- mials of degree ≤ k + 1 for both velocity and concentration, have been recently established in [M. Alvarez et al., M3AS: Math. Models Methods Appl. Sci. 26 (5) (2016) 867{900]. In this work we derive two efficient and reliable residual-based a posteriori error estimators for that scheme. For the first estimator we make use of suitable ellipticity and inf-sup conditions together with a Helmholtz decomposition and the local approximation properties of the Cl´ement interpolant and Raviart- Thomas operator to show its reliability, whereas the efficiency follows from inverse inequalities and localization arguments based on triangle-bubble and edge-bubble functions. Next, we analyze an alternative error estimator, whose reliability can be proved without resorting to Helmholtz decom- positions. Finally, we provide some numerical results confirming the reliability and efficiency of the estimators and illustrating the good performance of the associated adaptive algorithm for the augmented mixed-primal finite element method. Key words: Brinkman-transport coupling, nonlinear advection-diffusion, augmented mixed{primal formulation, sedimentation-consolidation process, finite element methods, a posteriori error analysis Mathematics subject classifications (2000): 65N30, 65N12, 76R05, 76D07, 65N15. 1 Introduction We have recently analyzed in [3], the solvability of a strongly coupled flow and transport system typically encountered in continuum-based models of sedimentation-consolidation processes. More ∗This work was partially supported by CONICYT-Chile through BASAL project CMM, Universidad de Chile, and project Anillo ACT1118 (ANANUM); by the Ministery of Education through the project REDOC.CTA of the Graduate School, Universidad de Concepci´on;and by Centro de Investigaci´onen Ingenier´ıaMatem´atica(CI2MA), Universidad de Concepci´on. ySecci´onde Matem´atica,Sede de Occidente, Universidad de Costa Rica, San Ram´onde Alajuela, Costa Rica, email: [email protected]. Present address: CI2MA and Departamento de Ingenier´ıaMatem´atica,Universi- dad de Concepci´on,Casilla 160-C, Concepci´on,Chile, email: [email protected]. zCI2MA and Departamento de Ingenier´ıaMatem´atica, Universidad de Concepci´on, Casilla 160-C, Concepci´on,Chile, email: [email protected]. xMathematical Institute, Oxford University, Andrew Wiles Building, Woodstock Road, Oxford OX2 6GG, UK, email: [email protected]. 1 precisely, the steady-state regime of a solid-liquid suspension immersed in a viscous fluid within a permeable medium is considered in [3], and the governing equations consist in the Brinkman problem with variable viscosity coupled with a nonlinear advection { nonlinear diffusion equation describing the transport model. An augmented variational formulation is then proposed there with main un- knowns given by the Cauchy pseudo-stress and bulk velocity of the mixture, and the solids volume 1 1 fraction, which are sought in H(div; Ω), H (Ω), and H (Ω), respectively. Fixed point arguments, cer- tain regularity assumptions, and some classical results concerning variational problems and Sobolev spaces are combined to establish the solvability of the continuous and discrete coupled formulations. Consequently, the rows of the Cauchy stress tensor were approximated with Raviart-Thomas elements of order k, whereas the velocity and solids concentration were discretized with continuous piecewise polynomials of degree ≤ k + 1. Suitable Strang-type estimates are employed to derive optimal a priori error estimates for the Galerkin solution. The purpose of this work is to provide reliable and efficient residual-based a posteriori error esti- mators for the steady sedimentation-consolidation system studied in [3]. Estimators of this kind are frequently employed to guide adaptive mesh refinement in order to guarantee an adequate convergence behavior of the Galerkin approximations, even under the eventual presence of singularities. The global estimator η depends on local estimators ηT defined on each element T of a given mesh Th. Then, η is said to be efficient (resp. reliable) if there exists a constant Ceff > 0 (resp. Crel > 0), independent of meshsizes, such that Ceff η + h.o.t. ≤ kerrork ≤ Crel η + h.o.t. ; where h.o.t. is a generic expression denoting one or several terms of higher order. Up to the authors knowledge, a number of a posteriori error estimators specifically targeted for non-viscous flow (e.g, Darcy) with transport problems are available in the literature [7, 15, 26, 32, 35]. However, only [8, 27] and [4] are devoted to the a posteriori error analysis for coupled viscous flow-transport problems. In particular, we derive in [4], two efficient and reliable residual-based a posteriori error estimators for an augmented mixed{primal finite element approximation of a stationary viscous flow and trans- port problem, which serves as a prototype model for sedimentation-consolidation processes and other phenomena where the transport of species concentration within a viscous fluid is of interest. In this paper, as well as in [4], we make use of ellipticity and inf-sup conditions together with a Helmholtz decomposition, local approximation properties of the Cl´ement interpolant and Raviart- Thomas operator, and known estimates from [6], [17], [21], [23] and [24], to prove the reliability of a residual-based estimator. Then, inverse inequalities, the localization technique based on triangle- bubble and edge-bubble functions imply the efficiency of the estimator. Alternatively, we deduce a second reliable and efficient residual-based a posteriori error estimator, where the Helmholtz decom- position is not employed in the corresponding proof of reliability. The rest of this paper is organized as follows. In Section2, we first recall from [3] the model problem and a corresponding augmented mixed-primal formulation as well as the associated Galerkin scheme. In Section3, we derive a reliable and efficient residual-based a posteriori error estimator for our Galerkin scheme. A second estimator is introduced and analyzed in Section4. Next, the analysis and results from Section3 and4 are extend to the three-dimensional case in Section5. Finally, in Section6, our theoretical results are illustrated via some numerical examples, highlighting also the good performance of the scheme and properties of the proposed error indicators. 2 The sedimentation-consolidation system n Let us denote by Ω ⊆ R , n = 2; 3 a given bounded domain with polyhedral boundary Γ = Γ¯D [ Γ¯N, with ΓD \ΓN = ; and jΓDj; jΓNj > 0, and denote by ν the outward unit normal vector on Γ. Standard 2 p s notation will be adopted for Lebesgue spaces L (Ω) and Sobolev spaces H (Ω) with norm k·ks;Ω and 1=2 1 −1=2 seminorm j · js;Ω. In particular, H (Γ) is the space of traces of functions of H (Ω) and H (Γ) denotes its dual. By M; M we will denote the corresponding vectorial and tensorial counterparts of the generic scalar functional space M. We recall that the space 2 2 H(div; Ω) := fτ 2 L (Ω) : div τ 2 L (Ω)g ; equipped with the usual norm 2 2 2 kτ kdiv;Ω := kτ k0;Ω + kdiv τ k0;Ω n×n is a Hilbert space. As usual, I stands for the identity tensor in R , and j · j denotes both the Euclidean norm in Rn and the Frobenius norm in Rn×n. 2.1 The governing equations The following model describes the steady state of the sedimentation-consolidation process consisting on the transport and suspension of a solid phase into an immiscible fluid contained in a vessel Ω (cf. [3]). The flow patterns are influenced by gravity and by the local fluctuations of the solids volume fraction. After elimination of the fluid pressure (cf. [3]), the process is governed by the following system of partial differential equations: 1 σd = ru ; K−1u − div σ = fφ ; div u = 0 in Ω ; µ(φ) (2.1) σe = #(φ) rφ − φu − fbk(φ)k ; β φ − div σe = g in Ω ; along with the following boundary conditions: u = uD on ΓD ; σν = 0 on ΓN ; (2.2) φ = φD on ΓD ; and σe · ν = 0 on ΓN ; where (·)d denotes the deviatoric operator. The sought quantities are the Cauchy fluid pseudo-stress σ, the average velocity of the mixture u, and the volumetric fraction of the solids (in short, concentration) φ. In this context, the parameter β is a positive constant representing the porosity of the medium, n×n and the permeability tensor K 2 C(Ω)¯ := [C(Ω)]¯ and its inverse are symmetric and uniformly positive definite, which means that there exists αK > 0 such that t −1 2 n v K (x)v ≥ αK jvj 8 v 2 R ; 8 x 2 Ω: Here, we assume that the kinematic effective viscosity, µ; the one-directional Kynch batch flux density function describing hindered settling, fbk; and the diffusion or sediment compressibility, #; are nonlin- ear
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