Least Squares Finite Element Methods

Abstract

Abstract. Least-squares finite element methods are an attractive class of methods for the numerical solution of partial differential equations. They are motivated by the desire to recover, in general settings, the advantageous features of Rayleigh-Ritz methods such as the avoidance of discrete compatibility conditions and the production of symmetric and positive definite discrete systems. The methods are based on the minimization of convex functionals that are constructed from equation residuals. This paper focuses on theoretical and practical aspects of least-square finite element methods and includes discussions of what issues enter into their construction, analysis, and performance. It also includes a discussion of some open problems.

Galerkin-least square B-spline approach toward advection–diffusion equation

“…This error is preserved by the Galerkin finite element approximation as well [21]. Now we notice from [19] that if w h is a least square minimizer and defined by a polynomial of degree less or equal k then…”

“…Now in general, if we have r + 1 data points then there is exactly one polynomial of degree at most r passing through the data points and the error in the interpolating polynomial is proportional to a power of the distance between the data points. A detailed discussion about the polynomial approximation and least squares piecewise polynomials approximations can be found in [19][20][21].…”

“…For stability analysis, we substitute σ n m = ω n e iθ mh ; (i = √ −1) in (19) which gives us growth factor ω of the form…”

Nonconforming Least-Squares Spectral Element Method for European Options

“…Since, our method is a Least-Squares method [17], we use the preconditioned conjugate gradient method (PCGM) for solving the Normal Equations. Now from [16] …”

A priori and a posteriori error analysis of a mixed scheme for the Brinkman problem

“…Several numerical methods exploit these properties, as for instance, different formulations based on least-squares, stabilization techniques, mixed finite elements, spectral discretizations, and hybridizable discontinuous Galerkin methods (see for instance [3,4,8,12,14,18,19,[21][22][23]26,[34][35][36], and the references therein). For the generalized Stokes problem written in velocity-vorticity-pressure variables, we mention [6] where an augmented mixed formulation based on RT k − P k+1 − P k+1 (with continuous pressure approximation) finite elements has been developed and analyzed.…”