Least-Squares FEM: Literature Review

Kayser‐Herold, Oliver; Matthies, Hermann G.

doi:10.24355/dbbs.084-200511080100-358

Cited by 3 publications

References 59 publications

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

Schwarz

Nickaeen

Serdas

et al. 2016

IJCSE

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In the present contribution we compare different mixed least-squares finite element formulations (LSFEMs) with respect to computational costs and accuracy. In detail, we consider an approach for Newtonian fluid flow, which is described by the incompressible Navier-Stokes equations. Starting from the residual forms of the equilibrium equation and the continuity condition, various first-order systems are derived. From these systems least-squares functionals are constructed by means of L 2-norms, which are the basis for the associated minimization problems. The first formulation under consideration is a div-grad first-order system resulting in a threefield formulation with stresses, velocities, and pressure as unknowns. This S-V-P formulation is approximated in H(div) × H 1 × L 2 on triangles and for comparison also in H 1 × H 1 × L 2 on quadrilaterals. The second formulation is the well-known div-curl-grad first-order velocityvorticity-pressure (V-V-P) formulation. Here all unknowns are approximated in H 1 on quadrilaterals. Besides some numerical advantages, as e.g. an inherent symmetric structure of the system of equations and a directly available error estimator, it is known that least-squares methods have also a drawback concerning mass conservation, especially when lower-order elements are used. Therefore, the main focus of the work is on performance and accuracy aspects on the one side for finite elements with different interpolation orders and on the other side on the usage of efficient solvers, for instance of Krylov-space or multigrid type. In order to demonstrate the capability of the formulations the results for some well-known benchmark problems are presented and discussed.

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

Schwarz

Nickaeen

Serdas

et al. 2016

IJCSE

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

Schwarz

Dwight

2018

Int. J. UncertaintyQuantification

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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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Least-Squares Methods for the Solution of Fluid-Structure Interaction Problems

Kayser‐Herold¹

2006

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Least-Squares FEM: Literature Review

Cited by 3 publications

References 59 publications

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

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

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

Least-Squares Methods for the Solution of Fluid-Structure Interaction Problems

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