• 제목/요약/키워드: fully discrete approximations

검색결과 12건 처리시간 0.019초

ERROR ESTIMATES OF FULLY DISCRETE DISCONTINUOUS GALERKIN APPROXIMATIONS FOR LINEAR SOBOLEV EQUATIONS

  • Ohm, M.R.;Shin, J.Y.;Lee, H.Y.
    • Journal of applied mathematics & informatics
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    • 제27권5_6호
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    • pp.1221-1234
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    • 2009
  • In this paper, we construct fully discrete discontinuous Galerkin approximations to the solution of linear Sobolev equations. We apply a symmetric interior penalty method which has an interior penalty term to compensate the continuity on the edges of interelements. The optimal convergence of the fully discrete discontinuous Galerkin approximations in ${\ell}^{\infty}(L^2)$ norm is proved.

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L2-ERROR ANALYSIS OF FULLY DISCRETE DISCONTINUOUS GALERKIN APPROXIMATIONS FOR NONLINEAR SOBOLEV EQUATIONS

  • Ohm, Mi-Ray;Lee, Hyun-Young
    • 대한수학회보
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    • 제48권5호
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    • pp.897-915
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    • 2011
  • In this paper, we develop a symmetric Galerkin method with interior penalty terms to construct fully discrete approximations of the solution for nonlinear Sobolev equations. To analyze the convergence of discontinuous Galerkin approximations, we introduce an appropriate projection and derive the optimal $L^2$ error estimates.

OPTIMAL L2-ERROR ESTIMATES FOR EXPANDED MIXED FINITE ELEMENT METHODS OF SEMILINEAR SOBOLEV EQUATIONS

  • Ohm, Mi Ray;Lee, Hyun Young;Shin, Jun Yong
    • 대한수학회지
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    • 제51권3호
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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.

ERROR ESTIMATES FOR FULLY DISCRETE DISCONTINUOUS GALERKIN METHOD FOR NONLINEAR PARABOLIC EQUATIONS

  • Ohm, Mi-Ray;Lee, Hyun-Yong;Shin, Jun-Yong
    • Journal of applied mathematics & informatics
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    • 제28권3_4호
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    • pp.953-966
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    • 2010
  • In this paper, we develop discontinuous Galerkin methods with penalty terms, namaly symmetric interior penalty Galerkin methods to solve nonlinear parabolic equations. By introducing an appropriate projection of u onto finite element spaces, we prove the optimal convergence of the fully discrete discontinuous Galerkin approximations in ${\ell}^2(L^2)$ normed space.

ERROR ESTIMATES FOR FULLY DISCRETE MIXED DISCONTINUOUS GALERKIN APPROXIMATIONS FOR PARABOLIC PROBLEMS

  • OHM, MI RAY;LEE, HYUN YOUNG;SHIN, JUN YONG
    • East Asian mathematical journal
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    • 제31권5호
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    • pp.685-693
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    • 2015
  • In this paper, we introduce fully discrete mixed discontinuous Galerkin approximations for parabolic problems. And we analyze the error estimates in $l^{\infty}(L^2)$ norm for the primary variable and the error estimates in the energy norm for the primary variable and the flux variable.

A PRIORI $L^2$-ERROR ESTIMATES OF THE CRANK-NICOLSON DISCONTINUOUS GALERKIN APPROXIMATIONS FOR NONLINEAR PARABOLIC EQUATIONS

  • Ahn, Min-Jung;Lee, Min-A
    • East Asian mathematical journal
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    • 제26권5호
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    • pp.615-626
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    • 2010
  • In this paper, we analyze discontinuous Galerkin methods with penalty terms, namly symmetric interior penalty Galerkin methods, to solve nonlinear parabolic equations. We construct finite element spaces on which we develop fully discrete approximations using extrapolated Crank-Nicolson method. We adopt an appropriate elliptic-type projection, which leads to optimal ${\ell}^{\infty}$ ($L^2$) error estimates of discontinuous Galerkin approximations in both spatial direction and temporal direction.

ERROR ANALYSIS OF FINITE ELEMENT APPROXIMATION OF A STEFAN PROBLEM WITH NONLINEAR FREE BOUNDARY CONDITION

  • Lee H.Y.
    • Journal of applied mathematics & informatics
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    • 제22권1_2호
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    • pp.223-235
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    • 2006
  • By applying the Landau-type transformation, we transform a Stefan problem with nonlinear free boundary condition into a system consisting of a parabolic equation and the ordinary differential equations. Fully discrete finite element method is developed to approximate the solution of a system of a parabolic equation and the ordinary differential equations. We derive optimal orders of convergence of fully discrete approximations in $L_2,\;H^1$ and $H^2$ normed spaces.

FULLY DISCRETE MIXED FINITE ELEMENT METHOD FOR A QUASILINEAR STEFAN PROBLEM WITH A FORCING TERM IN NON-DIVERGENCE FORM

  • Lee, H.Y.;Ohm, M.R.;Shin, J.Y.
    • Journal of applied mathematics & informatics
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    • 제24권1_2호
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    • pp.191-207
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    • 2007
  • Based on a mixed Galerkin approximation, we construct the fully discrete approximations of $U_y$ as well as U to a single-phase quasilinear Stefan problem with a forcing term in non-divergence form. We prove the optimal convergence of approximation to the solution {U, S} and the superconvergence of approximation to $U_y$.

ERROR ESTIMATE OF EXTRAPOLATED DISCONTINUOUS GALERKIN APPROXIMATIONS FOR THE VISCOELASTICITY TYPE EQUATION

  • Ohm, Mi-Ray;Lee, Hyun-Yong;Shin, Jun-Yong
    • Journal of applied mathematics & informatics
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    • 제29권1_2호
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    • pp.311-326
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    • 2011
  • In this paper, we adopt discontinuous Galerkin methods with penalty terms namely symmetric interior penalty Galerkin methods, to solve nonlinear viscoelasticity type equations. We construct finite element spaces and define an appropriate projection of u and prove its optimal convergence. We construct extrapolated fully discrete discontinuous Galerkin approximations for the viscoelasticity type equation and prove ${\ell}^{\infty}(L^2)$ optimal error estimates in both spatial direction and temporal direction.