We consider the application of least-squares variational principles to the numerical solution of partial differential equations. Our main focus is on the development of least-squares finite element methods for elliptic boundary value problems arising in fields such as fluid flows, linear elasticity, and convection-diffusion. For many of these problems, least-squares principles offer numerous theoretical and computational advantages in the algorithmic design and implementation of corresponding finite element methods that are not present in standard Galerkin discretizations. Most notably, the use of least-squares principles leads to symmetric and positive definite algebraic problems and allows us to circumvent stability conditions such as the inf-sup condition arising in mixed methods for the Stokes and Navier--Stokes equations. As a result, application of least-squares principles has led to the development of robust and efficient finite element methods for a large class of problems of practical importance.

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Finite Element Methods of Least-Squares Type

This article introduces a high-accuracy discrete singular convolution (DSC) for the numerical simulation of coupled convective heat transfer problems. The problem of a buoyancy-driven cavity is solved by two completely independent numerical procedures. One is a quasi-wavelet-based DSC approach, which uses the regularized Shannon's kernel, while the other is a standard form of the Galerkin finite-element method. The integration of the Navier-Stokes and energy equations is performed by employing velocity correction-based schemes. The entire laminar natural convection range of 10 3 h Ra h 10 8 is numerically simulated by both schemes. The reliability and robustness of the present DSC approach is extensively tested and validated by means of grid sensitivity and convergence studies. As a result, a set of new benchmark quality data is presented. The study emphasizes quantitative, rather than qualitative comparisons.

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A new benchmark quality solution for the buoyancy-driven cavity by discrete singular convolution

https://ntrs.nasa.gov/api/citations/19930013636/downloads/19930013636.pdf

Large-scale computation of incompressible viscous flow by least-squares finite element method

The viscous linear stability of four classes of incompressible flows inside rectangular containers is studied numerically. In the first class the instability of flow through a rectangular duct, driven by a constant pressure gradient along the axis of the duct (essentially a two-dimensional counterpart to plane Poiseuille flow – PPF), is addressed. The other classes of flow examined are generated by tangential motion of one wall, in one case in the axial direction of the duct, in another perpendicular to this direction, corresponding respectively to the two-dimensional counterpart to plane Couette flow (PCF) and the classic lid-driven cavity (LDC) flow, and in the fourth case a combination of both the previous tangential wall motions. The partial-derivative eigenvalue problem which in each case governs the temporal development of global three-dimensional small-amplitude disturbances is solved numerically. The results of Tatsumi & Yoshimura (1990) for pressure-gradient-driven flow in a rectangular duct have been confirmed; the relationship between the eigenvalue spectrum of PPF and that of the rectangular duct has been investigated. Despite extensive numerical experimentation no unstable modes have been found in the wall-bounded Couette flow, this configuration found here to be more stable than its one-dimensional limit. In the square LDC flow results obtained are in line with the predictions of Ding & Kawahara (1998b), Theofilis (2000) and Albensoeder et al. (2001b) as far as one travelling unstable mode is concerned. However, in line with the predictions of the latter two works and contrary to all previously published results it is found that this mode is the third in significance from an instability analysis point of view. In a parameter range unexplored by Ding & Kawahara (1998b) and all prior investigations two additional eigenmodes exist, which are both more unstable than the mode that these authors discovered. The first of the new modes is stationary (and would consequently be impossible to detect using power-series analysis of experimental data), whilst the second is travelling, and has a critical Reynolds number and frequency well inside the experimentally observed bracket. The effect of variable aspect ratio $A\in[0.5,4]$ of the cavity on the most unstable eigenmodes is also considered, and it is found that an increase in aspect ratio results in general destabilization of the flow. Finally, a combination of wall-bounded Couette and LDC flow, generated in a square duct by lid motion at an angle $\phi\in(0,{\pi}/{2})$ with the homogeneous duct direction, is shown to be linearly unstable above a Reynolds number $\Rey\,{=}\,800$ (based on the lid velocity and the duct length/height) at all $\phi$ parameter values examined. The excellent agreement with experiment in LDC flow and the alleviation of the erroneous prediction of stability of wall-bounded Couette flow is thus attributed to the presence of in-plane basic flow velocity components.

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Viscous linear stability analysis of rectangular duct and cavity flows

The current state and future direction of Eulerian models in simulating the tropospheric chemistry and transport of trace species: a review

SUMMARY The time-dependent Navier-Stokes equations and the energy balance equation for an incompressible, constant property fluid in the Boussinesq approximation are solved by a least-squares finite element method based on a velocity-pressure-vorticity-temperature-heat-flux (u-P-w- T-q) formulation discretized by backward finite differencing in time. The discretization scheme leads to the minimization of the residual in the 12-norm for each time step. Isoparametric bilinear quadrilateral elements and reduced integration are employed. Three examples, thermally driven cavity flow at Rayleigh numbers up to lo6, lid-driven cavity flow at Reynolds numbers up to lo4 and flow over a square obstacle at Reynolds number 200, are presented to validate the method. The past decade has witnessed a great deal of progress in the area of computational fluid dynamics. Numerous flow problems have been successfully solved by finite difference, finite volume and finite element methods. Most finite element methods are based on the Galerkin method, 9 ' the Taylor-Galerkin method and the Petrov-Galerkin method. 3-5 Mixed-order interpolation and penalty approach are commonly used in these methods. It is well known that these methods often lead to large, sparse, unsymmetric linear systems which are difficult to solve numerically. This explains why finite element analysis for three-dimensional fluid flow problems is not a common practice. To overcome this difficulty, we propose and develop a Least-Squares Finite Element Method (LSFEM) for time-dependent incompressible flow problems. The linear systems resulting from the discretization of LSFEM are always symmetrical and positive-definite. Therefore, they can be solved more easily and efficiently. This is the main reason for investigating the least-squares finite element approach. Least-squares finite element methods have already been applied with some success to com- pressible Euler and hyperbolic equations. Jiang and Carey6- ' and Jiang and Povinelli' used an implicit method for compressible flows. To further the capabilities of the method, Lefebvre et al.' applied unstructured triangular meshes to compressible flow problems. For transient advection problems, Donea and Quartapelle lo classified four different LSFEM approaches: characteristic LSFEM proposed by Li," LSFEM by Carey and Jiang," Taylor LSFEM by Park and Liggett l3 and space-time LSFEM by Nguyen and Reynen.I4

A least‐squares finite element method for time‐dependent incompressible flows with thermal convection

SUMMARY A time-accurate least-squares finite element method is used to simulate three-dimensional flows in a cubic cavity with a uniform moving top. The time- accurate solutions are obtained by the Crank-Nicolson method for time integration and Newton linearization for the convective terms with extensive linearization steps. A matrix-free algorithm of the Jacobi conjugate gradient method is used to solve the symmetric, positive definite linear system of equations. To show that the least-squares finite element method with the Jacobi conjugate gradient technique has promising potential to provide implicit, fully coupled and time-accurate solutions to large-scale three-dimensional fluid flows, we present results for three-dimensional lid-driven flows in a cubic cavity for Reynolds numbers up to 3200.

Transient solutions for three‐dimensional lid‐driven cavity flows by a least‐squares finite element method

SUMMARY Numerical results for time-dependent 2D and 3D thermocapillary flows are presented in this work. The numerical algorithm is based on the Crank‐Nicolson scheme for time integration, Newton’s method for linearization, and a least-squares finite element method, together with a matrix-free Jacobi conjugate gradient technique. The main objective in this work is to demonstrate how the least-squares finite element method, together with an iterative procedure, deals with the capillary-traction boundary conditions at the free surface, which involves the coupling of velocity and temperature gradients. Mesh refinement studies were also carried out to validate the numerical results. © 1998 John Wiley & Sons, Ltd.

Simulations of 2d and 3d thermocapillary flows by a least-squares finite element method

We study the regularity of the viscosity solution to the fully nonlinear parabolic thin obstacle problem. In particular, we prove that the solution is local H1+α on each side of the smooth obstacle, for some small α > 0. Following the method which was first introduced for the harmonic case by Caffarelli in 1979, we extend the results of FernándezReal (2016) who treated the fully nonlinear elliptic case. Our results also extend those of Chatzigeorgiou (2019) in two ways. First, we do not assume solutions nor operators to be symmetric. Second, our estimates are local, in the sense that do not rely on the boundary data.

Li Q. Tang

Papers

A least‐squares finite element method for time‐dependent incompressible flows with thermal convection

Transient solutions for three‐dimensional lid‐driven cavity flows by a least‐squares finite element method

Simulations of 2d and 3d thermocapillary flows by a least-squares finite element method

$H^{1+\alpha}$ estimates for the fully nonlinear parabolic thin obstacle problem