Nonlinear preconditioned conjugate gradient and least-squares finite elements
Computer Methods in Applied Mechanics and Engineering
Least-square finite elements for Stokes problem
Computer Methods in Applied Mechanics and Engineering
Least-squares finite element method for fluid dynamics
Computer Methods in Applied Mechanics and Engineering
Computer Methods in Applied Mechanics and Engineering - Special edition on the 20th Anniversary
Accelerated convergence in Newton's method
SIAM Review
Least-squares methods for Stokes equations based on a discrete minus one inner product
Journal of Computational and Applied Mathematics - Special issue on TICAM symposium
Least-Squares Finite Element Method for the Stokes Problem with Zero Residual of Mass Conservation
SIAM Journal on Numerical Analysis
Analysis of Least-Squares Finite Element Methods for the Navier--Stokes Equations
SIAM Journal on Numerical Analysis
SIAM Journal on Numerical Analysis
Finite Element Methods of Least-Squares Type
SIAM Review
Analysis of Velocity-Flux Least-Squares Principles for the Navier--Stokes Equations: Part II
SIAM Journal on Numerical Analysis
Least-squares spectral elements applied to the Stokes problem
Journal of Computational Physics
A Least-Squares Spectral Element Formulation for the Stokes Problem
Journal of Scientific Computing
Analysis of a Discontinuous Least Squares Spectral Element Method
Journal of Scientific Computing
Spectral/hp least-squares finite element formulation for the Navier-Stokes equations
Journal of Computational Physics
Least-squares finite element models of two-dimensional compressible flows
Finite Elements in Analysis and Design
Journal of Computational Physics
Journal of Computational Physics
Journal of Computational Physics
Journal of Computational Physics
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In this paper we examine the roles of minimization and linearization in the least-squares finite element formulations of nonlinear boundary-values problems. The least-squares principle is based upon the minimization of the least-squares functional constructed via the sum of the squares of appropriate norms of the residuals of the partial differential equations (in the present case we consider L"2 norms). Since the least-squares method is independent of the discretization procedure and the solution scheme, the least-squares principle suggests that minimization should be performed prior to linearization, where linearization is employed in the context of either the Picard or Newton iterative solution procedures. However, in the least-squares finite element analysis of nonlinear boundary-value problems, it has become common practice in the literature to exchange the sequence of application of the minimization and linearization operations. The main purpose of this study is to provide a detailed assessment on how the finite element solution is affected when the order of application of these operators is interchanged. The assessment is performed mathematically, through an examination of the variational setting for the least-squares formulation of an abstract nonlinear boundary-value problem, and also computationally, through the numerical simulation of the least-squares finite element solutions of both a nonlinear form of the Poisson equation and also the incompressible Navier-Stokes equations. The assessment suggests that although the least-squares principle indicates that minimization should be performed prior to linearization, such an approach is often impractical and not necessary.