• Korean Journal of Mathematics
• /
• v.31 no.4
• /
• pp.419-431
• /
• 2023

In this paper, we propose an iterative method for a symmetric interior penalty Galerkin method for heterogeneous elliptic problems. The iterative method consists mainly of two parts based on a non-overlapping domain decomposition approach. One is an intermediate preconditioner constructed by understanding the properties of the discontinuous finite element functions and the other is a preconditioning related to the dual-primal finite element tearing and interconnecting (FETI-DP) methodology. Numerical results for the proposed method are presented, which demonstrate the performance of the iterative method in terms of various parameters associated with the elliptic model problem, the finite element discretization, and non-overlapping subdomain decomposition.

• Composites Research
• /
• v.17 no.4
• /
• pp.25-31
• /
• 2004

In this study, a refined beam analysis model has been developed for multi-celled composite blades with elliptic cross-sections. Reissner's semi-complimentary energy functional is introduced to describe the beam theory and also to deal with the mixed-nature of the formulation. The wail of elliptic sections is discretized into finite number of elements along the contour line and Gauss integration is applied to obtain the section properties. For each cell of the section, a total of four continuity conditions are used to impose proper constraints for the section. The theory is applied to single- and double-celled composite blades with elliptic cross-sections and is validated with detailed finite element analysis results.

• Journal of the Korean Society for Industrial and Applied Mathematics
• /
• v.12 no.2
• /
• pp.69-79
• /
• 2008

Multigrid methods finite element/finite volume methods and their convergence properties are reviewed in a general setting. Some early theoretical results in simple finite element methods in variational setting method are given and extension to nonnested-noninherited forms are presented. Finally, the parallel theory for nonconforming element[13] and for cell centered finite difference methods [15, 23] are discussed.

• Communications of the Korean Mathematical Society
• /
• v.12 no.1
• /
• pp.17-25
• /
• 1997

We estimate the genus g(N) of modular curve $X_0^0(N)$ and show that g(N) = 0 if and only if $1 \leq N \leq 5$.

• The Pure and Applied Mathematics
• /
• v.12 no.2 s.28
• /
• pp.153-159
• /
• 2005

We investigate the error estimates of the h and p versions of the finite element method for an elliptic problems. We present theoretical results showing the p version gives results which are not worse than those obtained by the h version in the finite element method.

• Journal of the Korean Society for Industrial and Applied Mathematics
• /
• v.6 no.2
• /
• pp.85-97
• /
• 2002

In this paper, finite volume element methods for nonlinear parabolic problems are proposed and analyzed. Optimal order error estimates in $W^{1,p}$ and $L_p$ are derived for $2{\leq}p{\leq}{\infty}$. In addition, superconvergence for the error between the approximation solution and the generalized elliptic projection of the exact solution (or and the finite element solution) is also obtained.

• Communications of the Korean Mathematical Society
• /
• v.20 no.3
• /
• pp.563-578
• /
• 2005

In [Z. Cai and B. C. Shin, SIAM J. Numer. Anal. 40 (2002), 307-318], we developed the discrete first-order system least squares method for the second-order elliptic boundary value problem by directly approximating $H(div){\cap}H(curl)-type$ space based on the Helmholtz decomposition. Under general assumptions, error estimates were established in the $L^2\;and\;H^1$ norms for the vector and scalar variables, respectively. Such error estimates are optimal with respect to the required regularity of the solution. In this paper, we study solution methods for solving the system of linear equations arising from the discretization of variational formulation which possesses discrete biharmonic term and focus on numerical results including the performances of multigrid preconditioners and the finite element accuracy.

• Structural Engineering and Mechanics
• /
• v.2 no.1
• /
• pp.35-48
• /
• 1994

A plasticity-based concrete model is proposed. The failure surface is elliptic in the ${\sigma}-{\tau}$ stress space. Independent hardening as well as softening is assumed in tension, compression, and shear. The nonlinear inelastic action initiates from the origin in the ${\sigma}-{\varepsilon}$(${\tau}-{\gamma}$) diagram. Several parameters are incorporated to control hardening/softening regions. The model is incorporated into a nonlinear finite element program along with other classical models. Several examples are solved and the results are compared with experimental data and other failure criteria. "Reasonable results" and stable solutions are obtained for different types of reinforced concrete oriented structures.

• Bulletin of the Korean Mathematical Society
• /
• v.51 no.1
• /
• pp.139-156
• /
• 2014

In this paper, we investigate the error estimates of a quadratic elliptic control problem with pointwise control constraints. The state and the co-state variables are approximated by the order k = 1 Raviart-Thomas mixed finite element and the control variable is discretized by piecewise linear but discontinuous functions. Approximations of order $h^{\frac{3}{2}}$ in the $L^2$-norm and order h in the $L^{\infty}$-norm for the control variable are proved.

• Journal of applied mathematics & informatics
• /
• v.32 no.5_6
• /
• pp.707-719
• /
• 2014

In this paper, we investigate a variational discretization approximation of elliptic optimal control problems with control constraints. The state and the co-state are approximated by piecewise linear functions, while the control is not directly discretized. By using some proper intermediate variables, we derive a second-order convergence in $L^2$-norm and superconvergence between the numerical solution and elliptic projection of the exact solution in $H^1$-norm or the gradient of the exact solution and recovery gradient in $L^2$-norm. Then we construct a posteriori error estimates by using the superconvergence results and do some numerical experiments to confirm our theoretical results.