• Bulletin of the Korean Mathematical Society
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• v.49 no.4
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• pp.829-850
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• 2012

In this paper, we adopt symmetric interior penalty discontinuous Galerkin (SIPG) methods to approximate the solution of nonlinear viscoelasticity-type equations. We construct finite element space which consists of piecewise continuous polynomials. We introduce an appropriate elliptic-type projection and prove its approximation properties. We construct semidiscrete discontinuous Galerkin approximations and prove the optimal convergence in $L^2$ normed space.

• Journal of the Korean Mathematical Society
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• v.51 no.3
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• pp.545-565
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• 2014

In this paper we derive a priori $L^{\infty}(L^2)$ error estimates for expanded mixed finite element formulations of semilinear Sobolev equations. This formulation expands the standard mixed formulation in the sense that three variables, the scalar unknown, the gradient and the flux are explicitly treated. Based on this method we construct finite element semidiscrete approximations and fully discrete approximations of the semilinear Sobolev equations. We prove the existence of semidiscrete approximations of u, $-{\nabla}u$ and $-{\nabla}u-{\nabla}u_t$ and obtain the optimal order error estimates in the $L^{\infty}(L^2)$ norm. And also we construct the fully discrete approximations and analyze the optimal convergence of the approximations in ${\ell}^{\infty}(L^2)$ norm. Finally we also provide the computational results.

• Journal of the Korean Mathematical Society
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• v.51 no.1
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• pp.67-86
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• 2014

Based on an expanded mixed finite element method, we consider the semidiscrete approximations of the solution u of the quasilinear pseudo-parabolic equation defined on ${\Omega}{\subset}R^d$, $1{\leq}d{\leq}3$. We construct the semidiscrete approximations of ${\nabla}u$ and $a(u){\nabla}u+b(u){\nabla}u_t$ as well as u and prove the existence of the semidiscrete approximations. And also we prove the optimal convergence of ${\nabla}u$ and $a(u){\nabla}u+b(u){\nabla}u_t$ as well as u in $L^2$ normed space.

• Journal of applied mathematics & informatics
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• v.16 no.1_2
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• pp.165-181
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• 2004

Based on Landau-type transformation, a Stefan problem with non-linear free boundary condition is transformed into a system consisting of parabolic equation and the ordinary differential equations. Semidiscrete approximations are constructed. Optimal orders of convergence of semidiscrete approximation in $L_2$, $H^1$ and $H^2$ normed spaces are derived.

• Journal of applied mathematics & informatics
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• v.11 no.1_2
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• pp.185-199
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• 2003

In this paper, we apply finite element Galerkin method to a single-phase quasi-linear Stefan problem with a forcing term. We consider the existence and uniqueness of a semidiscrete approximation and optimal error estimates in $L_2$, $L_{\infty}$, $H_1$ and $H_2$ norms for semidiscrete Galerkin approximations we derived.

• Journal of applied mathematics & informatics
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• v.32 no.5_6
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• pp.585-598
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• 2014

In this paper we consider the nonlinear parabolic problems with mixed boundary condition. Under comparatively mild conditions of the coefficients related to the problem, we construct the discontinuous Galerkin approximation of the solution to the nonlinear parabolic problem. We discretize spatial variables and construct the finite element spaces consisting of discontinuous piecewise polynomials of which the semidiscrete approximations are composed. We present the proof of the convergence of the semidiscrete approximations in $L^{\infty}(H^1)$ and $L^{\infty}(L^2)$ normed spaces.

• Bulletin of the Korean Mathematical Society
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• v.52 no.5
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• pp.1445-1465
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• 2015

In this paper, the long time behavior of the convective Cahn-Hilliard equation in two dimensions is considered, semidiscrete and completely discrete spectral approximations are constructed, error estimates of optimal order that hold uniformly on the unbounded time interval $0{\leq}t<{\infty}$ are obtained.

• Journal of applied mathematics & informatics
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• v.15 no.1_2
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• pp.251-264
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• 2004

We introduce a semidiscrete mixed finite element approximation for the single-phase linear Stefan problem and show the unique existence of the approximation. And the optimal rate of convergence in $L^2$ and $H^1$ norms are derived.

• Journal of the Korean Society of Industry Convergence
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• v.12 no.3
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• pp.149-155
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• 2009

A new finite element equation is derived by applying quadratic and cubic time integration scheme to the variational formulation in time-integral for the analysis of the transient elastodynamic problems to increase the numerical accuracy and stability. Emphasis is focused on methodology for cubic time integration scheme procedure which are never presented before. In this semidiscrete approximations of the field variables, the time axis is divided equally and quadratic and cubic time variation is assumed in those intervals, and space is approximated by the usual finite element discretization technique. It is found that unconditionally stable numerical results are obtained in case of the cubic time variation. Some numerical examples are given to show the versatility of the presented formulation.