This paper introduces a genetic algorithm enhanced with a trap escaping strategy derived from the dual information presented as discrete Lagrange multipliers. When the genetic algorithm is trapped into a local optima, the Discrete Lagrange Multiplier method is called for the best individual found. The information provided by the Lagrangian method is unified, in the form of recombination, with t...

We obtain error estimates for finite element approximations of the lowest degree valid uniformly for a class of three-dimensional narrow elements. First, for the Lagrange interpolation we prove optimal error estimates, both in order and regularity, in Lp for p > 2. For p = 2 it is known that this result is not true. Applying extrapolation results we obtain an optimal order error estimate for fu...

In this paper we analyze the rate of local convergence of the augmented Lagrange method for solving optimization problems with equality constraints and the objective function expressed as the sum of a convex function and a twice continuously differentiable function. The presence of the non-smoothness of the convex function in the objective requires extensive tools such as the second-order varia...

Quadrature convergence of the extended Lagrange interpolant L2n+1f for any continuous function f is studied, where the interpolation nodes are the n zeros τi of an orthogonal polynomial of degree n and the n+ 1 zeros τ̂j of the corresponding “induced” orthogonal polynomial of degree n + 1. It is found that, unlike convergence in the mean, quadrature convergence does hold for all four Chebyshev w...

Abstract. We consider the finite element approximation of the Laplacian operator with the homogeneous Dirichlet boundary condition, and study the corresponding Lagrange interpolation in the context of finite element superconvergence. For ddimensional Qk-type elements with d ≥ 1 and k ≥ 1, we prove that the interpolation points must be the Lobatto points if the Lagrange interpolation and the fin...

The isogeometric analysis is increasingly used in various engineering problems. It is based on Non-Uniform Rational B-Splines (NURBS) basis function applied for the solution field approximation and the geometry description. One of the major concerns with this method is finding an efficient approach to impose essential boundary conditions, especially for inhomogeneous boundaries. The main contri...

In the Fictitious Domain Method with Lagrange multiplier (FDM) the physical domain is embedded into a simpler but larger domain called the fictitious domain. The partial differential equation is extended to the fictitious domain using a Lagrange multiplier to enforce the prescribed boundary conditions on the physical domain while all the other data are extended to the fictitious domain. This le...

A Schwarz Waveform Relaxation (SWR) algorithm is proposed to solve by Domain Decomposition Method (DDM) linear and nonlinear Schrödinger equations. The symbols of the transparent fractional transmission operators involved in Optimized Schwarz Waveform Relaxation (OSWR) algorithms are approximated by low order Lagrange polynomials to derive Lagrange-Schwarz Waveform Relaxation (LSWR) algorithms ...