First-order system least squares for second-order partial differential equations: part I
SIAM Journal on Numerical Analysis - Special issue: the articles in this issue are dedicated to Seymour V. Parter
First-Order System Least Squares for Second-Order Partial Differential Equations: Part II
SIAM Journal on Numerical Analysis
Finite Element Methods of Least-Squares Type
SIAM Review
Steepest descent using smoothed gradients
Applied Mathematics and Computation
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 Spectral Collocation for the Navier–Stokes Equations
Journal of Scientific Computing
Journal of Computational Physics
Least-Squares Finite Element Methods and Algebraic Multigrid Solvers for Linear Hyperbolic PDEs
SIAM Journal on Scientific Computing
Spectral collocation schemes on the unit disc
Journal of Computational Physics
SIAM Journal on Scientific Computing
International Journal of Numerical Modelling: Electronic Networks, Devices and Fields
Mass- and Momentum Conservation of the Least-Squares Spectral Element Method for the Stokes Problem
Journal of Scientific Computing
Journal of Scientific Computing
Journal of Computational Physics
Mathematics and Computers in Simulation
Journal of Computational Physics
Journal of Computational and Applied Mathematics
Computers & Mathematics with Applications
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This paper describes an equivalent but improved least-squares formulation for the numerical approximation of the solution of partial differential equations. Instead of using variational analysis to impose the conditions for minimizing the residual, the residuals are minimized directly, thus leading to a method we will denote by Direct Minimization (DM). DM circumvents setting up the normal equations which consists of matrix-matrix multiplications. Matrix-matrix multiplications are expensive, may lead to loss of accuracy and destroy the sparsity pattern present in the original system. The condition number of the DM formulation is the square root of the condition number which would be obtained if variational analysis was employed. An element-by-element procedure will be presented which allows for parallelization of DM. A computational comparison between DM and the conventional least-squares formulation based on variational analysis will be presented.