A comparative study of mixed least-squares FEMs for the incompressible Navier-Stokes equations

We investigate theoretically and numerically the use of the Least-Squares Finite-element method (LSFEM) to approach data-assimilation problems for the steady-state, incompressible Navier-Stokes equations. Our LSFEM discretization is based on a stressvelocity-pressure (S-V-P) first-order formulation, using discrete counterparts of the Sobolev spaces H(div) × H 1 × L 2 for the variables respectively. In general S-V-P formulations are promising when the stresses are of special interest, e.g. for non-Newtonian, multiphase or turbulent flows. Resolution of the system is via minimization of a leastsquares functional representing the magnitude of the residual of the equations. A simple and immediate approach to extend this solver to data-assimilation is to add a datadiscrepancy term to the functional. Whereas most data-assimilation techniques require a large number of evaluations of the forward-simulation and are therefore very expensive, the approach proposed in this work uniquely has the same cost as a single forward run. However, the question arises: what is the statistical model implied by this choice? We answer this within the Bayesian framework, establishing the latent background covariance model and the likelihood. Further we demonstrate that -in the linear case -the method is equivalent to application of the Kalman filter, and derive the posterior covariance. We practically demonstrate the capabilities of our method on a backward-facing step case. Our LSFEM formulation (without data) is shown to have good approximation quality, even on relatively coarse meshes -in particular with respect to mass-conservation and reattachment location. Adding limited velocity measurements from experiment, we show that the method is able to correct for discretization error on very coarse meshes, as well as correct for the influence of unknown and uncertain boundary-conditions.

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“…where we have used some additional physically motivated weights on the 1st and 2nd terms, see e.g. [40].…”

Section: Navier-stokes In σ − V − P Formmentioning

confidence: 99%

“…In this work we use the interpolation combination RT 1 P 3 P 1 , which seems to be a sufficient choice. For details about the implementation and other interpolation combinations the reader is referred to [40]. Finally, we obtain the system of equations for a typically element…”

Section: Navier-stokes In σ − V − P Formmentioning

confidence: 99%

Data Assimilation for Navier-Stokes Using the Least-Squares Finite-Element Method

Schwarz

Dwight

2018

Int. J. UncertaintyQuantification

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“…Arqub (2018) studied several classes of Robin time-fractional PDEs by reproducing kernel algorithm. Many numerical studies are available in the literature for solving integer-order problems, such as electromagnetic field problem (Zhao, 2013), inverse parabolic problem (Pourgholi et al, 2015), Benjamin-Bona-Mahony equation (Yu et al, 2016), Navier-Stokes equations (Schwarz et al, 2018) and nonlinear integer bilevel programming problems (Liu et al, 2020). A great deal of effort has been spent on constructing the analytical and numerical solutions for various fractional differential equations such as time-fractional mobile-immobile advection-dispersion equation (Kanth and Sirswal, 2018a), time-fractional diffusion equation (Kanth and Sirswal, 2018b), fractional discrete KdV equations (Kumar and Kumar, 2018), fractional Keller-Segel chemotaxis model (Morales-Delgado et al, 2018), time-fractional nonlinear dispersive PDEs (Ajou et al, 2019), fractional Kaup-Kupershmidt equation (Goufo et al, 2020), new Yang-Abdel-Aty-Cattani fractional diffusion equation (Kumar et al, 2020a), fractional immunogenetic tumour model (Ghanbari et al, 2020), fractional Lotka-Volterra population model (Kumar et al, 2020b), multidimensional time-fractional partial differential equation (Jleli et al, 2020) and delay fractional optimal control problems (Kheyrinataj and Nazemi, 2020).…”

Section: Introductionmentioning

confidence: 99%

Numerical treatment and analysis for a class of time-fractional Burgers equations with the Dirichlet boundary conditions

Kanth¹,

Garg²

2022

IJCSE

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This paper aims to study a class of time-fractional Burgers equations with the Dirichlet boundary conditions in the Caputo sense. The Burgers equation occurs in the study of fluid dynamics, turbulent flows, acoustic waves, and heat conduction. We discretise the equation by employing the Crank-Nicolson finite difference quadrature formula in the direction of time. We then discretise the resulting equations in space domain using the exponential B-splines. A rigorous study of stability and convergence analysis is analysed. Several test problems are studied to illustrate the efficacy and feasibility of the proposed method. Numerical simulations confirm the coherence with the theoretical analysis. Comparisons with the other existing results in the literature indicate the effectiveness of the method.

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“…Furthermore, the construction of the finite elements leads to positive definite and symmetric system matrices, also for differential equations with non self-adjoint operators and is applied successfully especially in fluid mechanics, see e.g. [1,4,5]. Besides, the LSFEM offers a wide range of approaches based on different solution variables, since they can be included directly in the formulations, which is a main point in the proposed approach.…”

Section: Introductionmentioning

confidence: 99%

A fluid‐structure interaction approach with inherently fulfilled coupling conditions

Schwarz

Averweg

Nisters

et al. 2018

Proc Appl Math and Mech

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In the present contribution a fluid-structure interaction (FSI) approach based on a mixed least-squares finite element method (LSFEM) is introduced. The LSFEM is a variational approach for constructing finite element formulations, see e.g.[1] and [2]. In this work, the proposed FSI method is based on the formulation of mixed finite elements in terms of stresses and velocities for both the fluid and the solid regime. The conforming discretization of the stresses and velocities in the spaces H(div) and H 1 , respectively, leads to the inherent fulfillment of the coupling conditions of a FSI method. The idea of the LSFEM based FSI scheme is the solution of the full problem consisting of both domains in one triangulation at the same time. The general idea on this was firstly annotated in [8]. A numerical example considering an incompressible, linear elastic solid at small deformations (compare to [3]) together with an incompressible laminar fluid flow demonstrates the applicability of the LSFEM-FSI approach.

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A comparative study of mixed least-squares FEMs for the incompressible Navier-Stokes equations

Cited by 4 publications

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Data Assimilation for Navier-Stokes Using the Least-Squares Finite-Element Method

Data Assimilation for Navier-Stokes Using the Least-Squares Finite-Element Method

Numerical treatment and analysis for a class of time-fractional Burgers equations with the Dirichlet boundary conditions

A fluid‐structure interaction approach with inherently fulfilled coupling conditions

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