Suh-Yuh Yang

A least-squares finite element method for incompressible flow in stress-velocity-pressure version☆

In this paper we are concerned with the incompressible flow in 2-D. Introducing additional variables of derivatives of velocity, which are called stresses here, the second-order dynamic equations are reduced into a first-order system with variables of stress, velocity and pressure. Combining the compatability conditions: and the divergence ice condition, we have a system with six first-order equations and six unknowns. Least-squares method performed over this extended system. The analysis shows that this method achieves optimal rates of convergence in the H1-norm as the h approaches to zero. Numerical experiences are also available.

Analysis of the L2 least-squares finite element method for the velocity-vorticity-pressure Stokes equations with velocity boundary conditions

A theoretical analysis of the L^2 least-squares finite element method (LSFEM) for solving the Stokes equations in the velocity-vorticity-pressure (VVP) first-order system formulation with the Dirichlet velocity boundary condition is given. The least-squares energy functional is defined to be the sum of the squared L^2-norms of the residuals in the partial differential equations, weighted appropriately by the viscosity constant @n. It is shown that, with many advantages, the method is stable and convergent without requiring extra smoothness of the exact solution, and the piecewise linear finite elements can be used to approximate all the unknowns. Furthermore, with respect to the order of approximation for smooth exact solutions, the method exhibits an optimal rate of convergence in the H^1-norm for velocity and a suboptimal rate of convergence in the L^2-norm for vorticity and pressure. Some numerical experiments in two and three dimensions are given, which confirm the a priori error estimates. Since the boundary of the bounded domain under consideration is polygonal in R^2 or polyhedral in R^3 instead of C^1-smooth, the authors adopt the more direct technique of Bramble-Pasciak and Cai-Manteuffel-McCormick, which departs from the Agmon-Douglis-Nirenberg theory, in showing the coercivity bound of the least-squares functional.

On camel-like traveling wave solutions in cellular neural networks

Abstract This paper is concerned with the existence of camel-like traveling wave solutions of cellular neural networks distributed in the one-dimensional integer lattice Z 1 . The dynamics of each given cell depends on itself and its nearest m left neighbor cells with instantaneous feedback. The profile equation of the infinite system of ordinary differential equations can be written as a functional differential equation in delayed type. Under appropriate assumptions, we can directly figure out the solution formula with many parameters. When the wave speed is negative and close to zero, we prove the existence of camel-like traveling waves for certain parameters. In addition, we also provide some numerical results for more general output functions and find out oscillating traveling waves numerically.

A bubble-stabilized least-squares finite element method for steady MHD duct flow problems at high Hartmann numbers

In this paper we devise a stabilized least-squares finite element method using the residual-free bubbles for solving the governing equations of steady magnetohydrodynamic duct flow. We convert the original system of second-order partial differential equations into a first-order system formulation by introducing two additional variables. Then the least-squares finite element method using C^0 linear elements enriched with the residual-free bubble functions for all unknowns is applied to obtain approximations to the first-order system. The most advantageous features of this approach are that the resulting linear system is symmetric and positive definite, and it is capable of resolving high gradients near the layer regions without refining the mesh. Thus, this approach is possible to obtain approximations consistent with the physical configuration of the problem even for high values of the Hartmann number. Before incoorperating the bubble functions into the global problem, we apply the Galerkin least-squares method to approximate the bubble functions that are exact solutions of the corresponding local problems on elements. Therefore, we indeed introduce a two-level finite element method consisting of a mesh for discretization and a submesh for approximating the computations of the residual-free bubble functions. Numerical results confirming theoretical findings are presented for several examples including the Shercliff problem.

SYNCHRONIZATION IN LINEARLY COUPLED DYNAMICAL NETWORKS WITH DISTRIBUTED TIME DELAYS

In this paper, we investigate the global exponential synchronization of linearly coupled dynamical networks with time delays. The time delay considered is of the distributed type and the outer-coupling matrix is not assumed to be symmetric. Employing the Lyapunov functional and matrix inequality techniques, we propose a sufficient condition for the occurrence of global exponential synchronization. Two illustrative examples, the coupled Chuas circuits and the coupled Hindmarsh–Rose neurons, and their numerical simulation results are presented to demonstrate the theoretical analyses.

Least-squares finite element methods for the elasticity problem

Abstract A new first-order system formulation for the linear elasticity problem in displacement-stress form is proposed. The formulation is derived by introducing additional variables of derivatives of the displacements, whose combinations represent the usual stresses. Standard and weighted least-squares finite element methods are then applied to this extended system. These methods offer certain advantages such as that they need not satisfy the inf-sup condition which is required in the mixed finite element formulation, that a single continuous piecewise polynomial space can be used for the approximation of all the unknowns, that the resulting algebraic systems are symmetric and positive definite, and that accurate approximations of the displacements and the stresses can be obtained simultaneously. With displacement boundary conditions, it is shown that both methods achieve optimal rates of convergence in the H 1 -norm and in the L 2 -norm for all the unknowns. Numerical experiments with various Poisson ratios are given to demonstrate the theoretical error estimates.

A Novel Least-Squares Finite Element Method Enriched with Residual-Free Bubbles for Solving Convection-Dominated Problems

In this paper we devise a novel least-squares finite element method (LSFEM) for solving scalar convection-dominated convection-diffusion problems. First, we convert a second-order convection-diffusion problem into a first-order system formulation by introducing the gradient

DIVERSITY OF TRAVELING WAVE SOLUTIONS IN DELAYED CELLULAR NEURAL NETWORKS

\mathbf{p}:=-\kappa\nabla u

Least-squares finite element approximations to the Timoshenko beam problem

as an additional variable. The LSFEM using continuous piecewise linear elements enriched with residual-free bubbles for both variables

Analysis of least squares finite element methods for a parameter-dependent first-order system

u