2D constrained Navier–Stokes equations

Modified augmented Lagrangian preconditioners for the incompressible NavierStokes equations

We study different variants of the augmented Lagrangian (AL)-based block-triangular preconditioner introduced by the first two authors in [SIAM J. Sci. Comput. 2006; 28: 2095–2113]. The preconditioners are used to accelerate the convergence of the Generalized Minimal Residual method (GMRES) applied to various finite element and Marker-and-Cell discretizations of the Oseen problem in two and thr...

A study on the global regularity for a model of the 3D axisymmetric NavierStokes equations

We investigates the global regularity issue concerning a model equation proposed by Hou and Lei [3] to understand the stabilizing effects of the nonlinear terms in the 3D axisymmetric Navier-Stokes and Euler equations. Two major results are obtained. The first one establishes the global regularity of a generalized version of their model with a fractional Laplacian when the fractional power sati...

‎A matrix LSQR algorithm for solving constrained linear operator equations

In this work‎, ‎an iterative method based on a matrix form of LSQR algorithm is constructed for solving the linear operator equation $mathcal{A}(X)=B$‎ ‎and the minimum Frobenius norm residual problem $||mathcal{A}(X)-B||_F$‎ ‎where $Xin mathcal{S}:={Xin textsf{R}^{ntimes n}~|~X=mathcal{G}(X)}$‎, ‎$mathcal{F}$ is the linear operator from $textsf{R}^{ntimes n}$ onto $textsf{R}^{rtimes s}$‎, ‎$ma...

Solving constrained Pell equations

Consider the system of Diophantine equations x2 − ay2 = b, P (x, y) = z2, where P is a given integer polynomial. Historically, such systems have been analyzed by using Baker’s method to produce an upper bound on the integer solutions. We present a general elementary approach, based on an idea of Cohn and the theory of the Pell equation, that solves many such systems. We apply the approach to th...

The 2D Incompressible Boussinesq Equations