A posteriori error analysis of an augmented mixed method for the Navier–Stokes equations with nonlinear viscosity

Numerical Methods Partial

Oyarzúa

et al. 2017

Self Cite

A new stress‐based mixed variational formulation for the stationary Navier‐Stokes equations with constant density and variable viscosity depending on the magnitude of the strain tensor, is proposed and analyzed in this work. Our approach is a natural extension of a technique applied in a recent paper by some of the authors to the same boundary value problem but with a viscosity that depends nonlinearly on the gradient of velocity instead of the strain tensor. In this case, and besides remarking that the strain‐dependence for the viscosity yields a more physically relevant model, we notice that to handle this nonlinearity we now need to incorporate not only the strain itself but also the vorticity as auxiliary unknowns. Furthermore, similarly as in that previous work, and aiming to deal with a suitable space for the velocity, the variational formulation is augmented with Galerkin‐type terms arising from the constitutive and equilibrium equations, the relations defining the two additional unknowns, and the Dirichlet boundary condition. In this way, and as the resulting augmented scheme can be rewritten as a fixed‐point operator equation, the classical Schauder and Banach theorems together with monotone operators theory are applied to derive the well‐posedness of the continuous and associated discrete schemes. In particular, we show that arbitrary finite element subspaces can be utilized for the latter, and then we derive optimal a priori error estimates along with the corresponding rates of convergence. Next, a reliable and efficient residual‐based a posteriori error estimator on arbitrary polygonal and polyhedral regions is proposed. The main tools used include Raviart‐Thomas and Clément interpolation operators, inverse and discrete inequalities, and the localization technique based on triangle‐bubble and edge‐bubble functions. Finally, several numerical essays illustrating the good performance of the method, confirming the reliability and efficiency of the a posteriori error estimator, and showing the desired behavior of the adaptive algorithm, are reported. © 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1692–1725, 2017

“…Finally, the derivation of the upper bound for

‖ {boldR}_{2} {false‖}_{ℍ_{0} (d i v; Ω)'}

makes use of a stable Helmholtz decomposition for

ℍ_{0} (d i v; Ω)

‖ {boldR}_{2} {false‖}_{ℍ_{0} (d i v; Ω)'}

it also appears the local term

h_{T}^{2} ‖ \nabla {bold-italicu}_{h} - {bold-italict}_{h} - {bold-italicρ}_{h} {false‖}_{0, T}^{2}

, which being dominated by

‖ e ({bold-italicu}_{h}) - {bold-italict}_{h} {false‖}_{0, T}^{2} + ‖ {bold-italicρ}_{h} - \nabla {bold-italicu}_{h} + e ({bold-italicu}_{h}) {false‖}_{0, T}^{2}

, is then omitted from the final definition of

Θ_{T}^{2}

(cf.…”

Section: A Posteriori Error Analysismentioning

confidence: 86%

bold-italict

and the presence of additional terms in the present augmented formulation, two of the resulting bounded functionals, whose norms need to be estimated later on, do not coincide with those in (see

{boldR}_{3}

and

{boldR}_{4}

below).Lemma Let

\overset{true\to}{bold-italict} \in H

and

{\overset{true\to}{bold-italict}}_{h} \in {boldH}_{h}

be the unique solutions of the continuous and discrete problems (3.5) and (4.1), respectively.…”

Section: A Posteriori Error Analysismentioning

confidence: 99%

“…More precisely, proceeding similarly as in , we begin with the result to be introduced next. However, we notice in advance that due to the new meaning of the unknown

bold-italict

and the presence of additional terms in the present augmented formulation, two of the resulting bounded functionals, whose norms need to be estimated later on, do not coincide with those in (see

{boldR}_{3}

and

{boldR}_{4}

below).Lemma Let

\overset{true\to}{bold-italict} \in H

and

{\overset{true\to}{bold-italict}}_{h} \in {boldH}_{h}

be the unique solutions of the continuous and discrete problems (3.5) and (4.1), respectively. In addition, given

ϵ \in (0, ϵ_{0})

, with

ϵ_{0}

defined by (3.18) (cf.…”

Section: A Posteriori Error Analysismentioning

confidence: 99%

“…In turn, to prove the efficiency of

\overset{Θ}{true}

it suffices to control the new terms, which is the purpose of the following two lemmas. Lemma There exist positive constants C 1 , C 2 , independent of h, such that a)

h_{T}^{2} ‖ c u r l ({bold-italict}_{h} + {bold-italicρ}_{h}) {false‖}_{0, T}^{2} \leq C_{1} {‖ t - {bold-italict}_{h} {false‖}_{0, T}^{2} + ‖ ρ - {bold-italicρ}_{h} {false‖}_{0, T}^{2}} \forall T \in {scriptT}_{h},

h_{e} {false| [​ ​ [({bold-italict}_{h} + {bold-italicρ}_{h}) s] ​ ​] false|}_{0, e}^{2} \leq C_{2} {‖ t - {bold-italict}_{h} {false‖}_{0, ω_{e}}^{2} + ‖ ρ - {bold-italicρ}_{h} {false‖}_{0, ω_{e}}^{2}} \forall e \in ℰ_{h} (Ω),

where

ω_{e} : = \cup {T' \in {scriptT}_{h} : e \in ℰ (T')}

.Proof For a) we refer to , Lemma 4.3] (see also , Lemma 3.15] or , Lemma 4.9]). Similarly, for b) we refer to , Lemma 4.4] (see also , Lemma 3.15] or , Lemma 4.10]).…”

Section: A Posteriori Error Analysismentioning

confidence: 99%

See 2 more Smart Citations

An augmented stress‐based mixed finite element method for the steady state Navier‐Stokes equations with nonlinear viscosity

Camaño

Numerical Methods Partial

Oyarzúa

et al. 2017

Self Cite

“…The remaining terms in 2 B,T and 2 D,T can be treated very much in the same way as done in [24,27,28], where the analysis is based on inverse inequalities found in [19], together with the localisation technique based on tetrahedron-bubble and facetbubble functions [34]. Such a theory requires further notation and preliminary results collected in what follows.…”

Section: Efficiencymentioning

confidence: 99%

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

Alvarez

Ruiz–Baier

2017

Calcolo

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 an augmented fully mixed formulation for the nonisothermal Oldroyd–Stokes problem

Caucao

Numerical Methods Partial

Oyarzúa

2018

Self Cite

In this article, we consider an augmented fully mixed variational formulation that has been recently proposed for the nonisothermal Oldroyd–Stokes problem, and develop an a posteriori error analysis for the 2‐D and 3‐D versions of the associated mixed finite element scheme. More precisely, we derive two reliable and efficient residual‐based a posteriori error estimators for this problem on arbitrary (convex or nonconvex) polygonal and polyhedral regions. The reliability of the proposed estimators draws mainly upon the uniform ellipticity of the bilinear forms of the continuous formulation, suitable assumptions on the domain and the data, stable Helmholtz decompositions, and the local approximation properties of the Clément and Raviart–Thomas operators. On the other hand, inverse inequalities, the localization technique based on bubble functions, and known results from previous works are the main tools yielding the efficiency estimate. Finally, several numerical results confirming the properties of the a posteriori error estimators and illustrating the performance of the associated adaptive algorithms are reported.