• Title/Summary/Keyword: fully discrete solution

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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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    • v.27 no.5_6
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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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ERROR ESTIMATES FOR A GALERKIN METHOD FOR A COUPLED NONLINEAR SCHRÖDINGER EQUATIONS

  • Omrani, Khaled;Rahmeni, Mohamed
    • Bulletin of the Korean Mathematical Society
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    • v.57 no.1
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    • pp.219-244
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    • 2020
  • In this paper, we approximate the solution of the coupled nonlinear Schrödinger equations by using a fully discrete finite element scheme based on the standard Galerkin method in space and implicit midpoint discretization in time. The proposed scheme guarantees the conservation of the total mass and the energy. First, a priori error estimates for the fully discrete Galerkin method is derived. Second, the existence of the approximated solution is proved by virtue of the Brouwer fixed point theorem. Moreover, the uniqueness of the solution is shown. Finally, convergence orders of the fully discrete Crank-Nicolson scheme are discussed. The end of the paper is devoted to some numerical experiments.

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

  • Ohm, Mi-Ray;Lee, Hyun-Young
    • Bulletin of the Korean Mathematical Society
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    • v.48 no.5
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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.

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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    • v.22 no.1_2
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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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    • v.24 no.1_2
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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 $U_y$.

A FINITE DIFFERENCE/FINITE VOLUME METHOD FOR SOLVING THE FRACTIONAL DIFFUSION WAVE EQUATION

  • Sun, Yinan;Zhang, Tie
    • Journal of the Korean Mathematical Society
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    • v.58 no.3
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    • pp.553-569
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    • 2021
  • In this paper, we present and analyze a fully discrete numerical method for solving the time-fractional diffusion wave equation: ∂βtu - div(a∇u) = f, 1 < β < 2. We first construct a difference formula to approximate ∂βtu by using an interpolation of derivative type. The truncation error of this formula is of O(△t2+δ-β)-order if function u(t) ∈ C2,δ[0, T] where 0 ≤ δ ≤ 1 is the Hölder continuity index. This error order can come up to O(△t3-β) if u(t) ∈ C3 [0, T]. Then, in combinination with the linear finite volume discretization on spatial domain, we give a fully discrete scheme for the fractional wave equation. We prove that the fully discrete scheme is unconditionally stable and the discrete solution admits the optimal error estimates in the H1-norm and L2-norm, respectively. Numerical examples are provided to verify the effectiveness of the proposed numerical method.

QUADRATURE BASED FINITE ELEMENT METHODS FOR LINEAR PARABOLIC INTERFACE PROBLEMS

  • Deka, Bhupen;Deka, Ram Charan
    • Bulletin of the Korean Mathematical Society
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    • v.51 no.3
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    • pp.717-737
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    • 2014
  • We study the effect of numerical quadrature in space on semidiscrete and fully discrete piecewise linear finite element methods for parabolic interface problems. Optimal $L^2(L^2)$ and $L^2(H^1)$ error estimates are shown to hold for semidiscrete problem under suitable regularity of the true solution in whole domain. Further, fully discrete scheme based on backward Euler method has also analyzed and optimal $L^2(L^2)$ norm error estimate is established. The error estimates are obtained for fitted finite element discretization based on straight interface triangles.