A priori and a posteriori error analyses of a pseudostress-based mixed formulation for linear elasticity

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

“…To this respect, we first observe that direct applications of the Cauchy‐Schwarz inequality give the corresponding estimates for the functionals

{boldR}_{1}, {boldR}_{3}

, and

{boldR}_{4}

. 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)

Section: A Posteriori Error Analysismentioning

confidence: 92%

“…The announced estimates are summarized in the following three lemmas.Lemma There exist positive constants C 5 , C 6 , independent of h, such that a)

h_{T}^{2} ‖ \underset{true_}{c u r l} ({bold-italict}_{h} + {bold-italicρ}_{h}) {false‖}_{0, T}^{2} \leq C_{5} {‖ 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}) \times ν] ​ ​] false|}_{0, e}^{2} \leq C_{6} {‖ 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 We refer to , Lemmas 4.9 and 4.10] for the proofs of a) and b).…”

Section: A Posteriori Error Analysismentioning

confidence: 99%

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

“…and 25) where c 3 > 0 is a constant depending on κ 3 , κ 4 , and the norm of the trace operator mapping…”

Section: Reliability Of the A Posteriori Error Estimatorsmentioning

confidence: 99%

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

Computers & Mathematics with Applications

Ruiz–Baier

Tierra

2016

Self Cite

In this work we develop the a posteriori error analysis of an augmented mixed finite element method for the 2D and 3D versions of the Navier-Stokes equations when the viscosity depends nonlinearly on the module of the velocity gradient. Two different reliable and efficient residual-based a posteriori error estimators for this problem on arbitrary (convex or non-convex) polygonal and polyhedral regions are derived. Our analysis of reliability of the proposed estimators draws mainly upon the global inf-sup condition satisfied by a suitable linearization of the continuous formulation, an application of Helmholtz decomposition, and the local approximation properties of the RaviartThomas and Clément interpolation operators. In addition, differently from previous approaches for augmented mixed formulations, the boundedness of the Clément operator plays now an interesting role in the reliability estimate. On the other hand, inverse and discrete inequalities, and the localization technique based on triangle-bubble and edge-bubble functions are utilized to show their efficiency. Finally, several numerical results are provided to illustrate the good performance of the augmented mixed method, to confirm the aforementioned properties of the a posteriori error estimators, and to show the behaviour of the associated adaptive algorithm.

“…We remark here that the 3D version of (5.12) and (5.13) is available in [17,Lemma 4.3]. Next, we let ζ := ∇z ∈ H 1 ( ), χ h := I h (χ ), and define…”

Section: Where (T ) and (F) Are The Union Of All Elements Intersectinmentioning

confidence: 99%

A Priori and a Posteriori Error Analyses of an Augmented HDG Method for a Class of Quasi-Newtonian Stokes Flows

Sequeira

2016

J Sci Comput

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In a recent work we developed a new hybridizable discontinuous Galerkin (HDG) method for a class of nonlinear Stokes models arising in quasi-Newtonian fluids. The approach there uses the incompressibility condition to eliminate the pressure, sets the gradient of the velocity as an auxiliary unknown, and enriches the original formulation with convenient redundant equations, thus giving rise to an augmented scheme. However, the corresponding analysis, which makes use of a fixed point strategy that depends on a suitably chosen parameter, yields optimal rates of convergence for only two of the six resulting unknowns, whereas the reported numerical results, showing higher orders than predicted, support the conjecture that the a priori error estimates are not sharp. In the present paper, the main features of the aforementioned augmented formulation are maintained, but after introducing slight modifications of the finite element subspaces for the pseudostress and velocity, we are able to significantly improve our previous analyses and results. More precisely, on one hand we realize here that it suffices to choose the stabilization tensor as the identity times the meshsize, and hence neither fixed-point arguments nor related parameters are needed anymore to establish the well-posedness of the discrete scheme, and on the other hand we now prove optimally convergent approximations for all the unknowns. Furthermore, we develop 123 J Sci Comput a reliable and efficient residual-based a posteriori error estimator, and propose the associated adaptive algorithm for our HDG approximation of the nonlinear model problem. Finally, several numerical results illustrating the performance of the method, confirming the theoretical properties of the estimator, and showing the expected behaviour of the adaptive refinements, are presented.