Byeong-Chun Shin

Some exact and new solutions of the Nizhnik-Novikov-Vesselov equation using the Exp-function method

In this paper, using the Exp-function method, we give some explicit formulas of exact traveling wave solutions for the Nizhnik-Novikov-Vesselov equation.

A comparison of the Newton–Krylov method with high order Newton-like methods to solve nonlinear systems

Abstract We compare the CPU time and error estimates of some variants of Newton method of the third and fourth-order convergence with those of the Newton–Krylov method used to solve systems of nonlinear equations. By expanding some numerical experiments we show that the use of Newton–Krylov method is better in the cost and accuracy points of view than the use of other high order Newton-like methods when the system is sparse and its size is large.

Solution Methods for the Poisson Equation with Corner Singularities: Numerical Results

In [Z. Cai and S. Kim, SIAM J. Numer. Anal., 39 (2001), pp. 286--299], we developed a new finite element method using singular functions for the Poisson equation on a polygonal domain with re-entrant angles. Such a method first computes the regular part of the solution, then the stress intensity factor, and finally the solution itself. This paper studies solution methods for solving the system of linear equations arising from the discretization and focuses on numerical results including the finite element accuracy and the multigrid performance.

The least-squares pseudo-spectral method for Navier-Stokes equations

A spectral collocation approximation of first-order system least squares for incompressible Stokes equations was analyzed in Kim et al. (2004) [12], and finite element approximations for incompressible Navier-Stokes equations were developed in Bochev et al. (1998,1999) [9,10]. The aim of this paper is to analyze the first-order system least-squares pseudo-spectral method for incompressible Navier-Stokes equations. The paper will be an extension of the result in Kim et al. (2004) [12] to the Navier-Stokes equations. Our least-squares functional is defined by the sum of discrete spectral norms of a first-order system of equations corresponding to the Navier-Stokes equations based on Legendre-Gauss-Lobatto points. We show its ellipticity and continuity over an appropriate product space, and spectral convergences of discretization errors are derived in the H^1-norm and the L^2-norm in each variable. Finally, we present some numerical examples.

PRECONDITIONED SPECTRAL COLLOCATION METHOD ON CURVED ELEMENT DOMAINS USING THE GORDON-HALL TRANSFORMATION

The spectral collocation method for a second order elliptic boundary value problem on a domain with curved boundaries is stud- ied using the Gordon and Hall transformation which enables us to have a transformed elliptic problem and a square domain S = (0,h) × (0,h), h > 0. The preconditioned system of the spectral collocation approx- imation based on Legendre-Gauss-Lobatto points by the matrix based on piecewise bilinear finite element discretizations is shown to have the high order accuracy of convergence and the efficiency of the finite element preconditioner.

The Discrete First-Order System Least Squares: The Second-Order Elliptic Boundary Value Problem

In [Z. Cai, T. Manteuffel, and S. F. McCormick, SIAM J. Numer. Anal., 34 (1997), pp. 425--454], an L2-norm version of first-order system least squares (FOSLS) was developed for scalar second-order elliptic partial differential equations. A limitation of this approach is the requirement of sufficient smoothness of the original problem, which is used for the equivalence of spaces between (H1)d and

Numerical Solution for Elliptic Interface Problems Using Spectral Element Collocation Method

H(\div)\cap H({\rm curl})

Newton's method for the Navier-Stokes equations with finite-element initial guess of stokes equations

-type, where d=2 or 3 is the dimension. By directly approximating

A least-squares/penalty method for distributed optimal control problems for Stokes equations

H(\div)\cap H({\rm curl})

Dynamics for controlled 2-D Boussinesq systems with distributed controls

-type space based on the Helmholtz decomposition, this paper develops a discrete FOSLS approach in two dimensions. Under general assumptions, we establish error estimates in the L2 and H1 norms for the vector and scalar variables, respectively. Such error estimates are optimal with respect to the required regularity of the solution. A preconditioner for the algebraic system arising from this approach is also considered.