• Title/Summary/Keyword: discontinuous Galerkin method

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hp-DISCONTINUOUS GALERKIN METHODS FOR THE LOTKA-MCKENDRICK EQUATION: A NUMERICAL STUDY

  • Jeong, Shin-Ja;Kim, Mi-Young;Selenge, Tsendanysh
    • Communications of the Korean Mathematical Society
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    • v.22 no.4
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    • pp.623-640
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    • 2007
  • The Lotka-McKendrick model which describes the evolution of a single population is developed from the well known Malthus model. In this paper, we introduce the Lotka-McKendrick model. We approximate the solution to the model using hp-discontinuous Galerkin finite element method. The numerical results show that the presented hp-discontinuous Galerkin method is very efficient in case that the solution has a sharp decay.

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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    • v.28 no.3_4
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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.

DEVELOPMENT OF HIGH-ORDER ADAPTIVE DISCONTINUOUS GALERKIN METHOD FOR UNSTEADY FLOW SIMULATION (비정상 유동 해석을 위한 고차정확도 격자 적응 불연속 갤러킨 기법 개발)

  • Lee, H.D.;Choi, J.H.;Kwon, O.J.
    • 한국전산유체공학회:학술대회논문집
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    • 2010.05a
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    • pp.534-541
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    • 2010
  • A high-order accurate Euler flow solver based on a discontinuous Galerkin method has been developed for the numerical simulation of unsteady flows on unstructured meshes. A multi-level solution-adaptive mesh refinement/coarsening technique was adopted to enhance the resolution of numerical solutions efficiently by increasing mesh density in the high-gradient region. An acoustic wave scattering problem was investigated to assess the accuracy of the present discontinuous Galerkin solver, and a supersonic flow in a wind tunnel with a forward facing step was simulated by using the adaptive mesh refinement technique. It was shown that the present discontinuous Galerkin flow solver can capture unsteady flows including the propagation and scattering of the acoustic waves as well as the strong shock waves.

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HIGHER ORDER DISCONTINUOUS GALERKIN FINITE ELEMENT METHODS FOR NONLINEAR PARABOLIC PROBLEMS

  • Ohm, Mi Ray;Lee, Hyun Young;Shin, Jun Yong
    • Journal of the Korean Society for Industrial and Applied Mathematics
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    • v.18 no.4
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    • pp.337-350
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    • 2014
  • In this paper, we consider discontinuous Galerkin finite element methods with interior penalty term to approximate the solution of nonlinear parabolic problems with mixed boundary conditions. We construct the finite element spaces of the piecewise polynomials on which we define fully discrete discontinuous Galerkin approximations using the Crank-Nicolson method. To analyze the error estimates, we construct an appropriate projection which allows us to obtain the optimal order of a priori ${\ell}^{\infty}(L^2)$ error estimates of discontinuous Galerkin approximations in both spatial and temporal directions.

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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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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    • v.26 no.5
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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.

DEVELOPMENT OF A HIGH-ORDER IMPLICIT DISCONTINUOUS GALERKIN METHOD FOR SOLVING COMPRESSIBLE NAVIER-STOKES EQUATIONS (압축성 Navier-Stokes 방정식 해를 위한 고차 정확도 내재적 불연속 갤러킨 기법의 개발)

  • Choi, J.H.;Lee, H.D.;Kwon, O.J.
    • Journal of computational fluids engineering
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    • v.16 no.4
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    • pp.72-83
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    • 2011
  • A high-order discontinuous Galerkin method for the two-dimensional compressible Navier-Stokes equations was developed on unstructured triangular meshes. For this purpose, the BR2 methd(the second Bassi and Rebay discretization) was adopted for space discretization and an implicit Euler backward method was used for time integration. Numerical tests were conducted to estimate the convergence order of the numerical solutions of the Poiseuille flow for which analytic solutions are available for comparison. Also, the flows around a flat plate, a 2-D circular cylinder, and an NACA0012 airfoil were numerically simulated. The numerical results showed that the present implicit discontinuous Galerkin method is an efficient method to obtain very accurate numerical solutions of the compressible Navier-Stokes equations on unstructured meshes.

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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    • v.29 no.1_2
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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.

DEVELOPMENT OF AN HIGH-ORDER IMPLICIT DISCONTINUOUS GALERKIN METHOD ON UNSTRUCTURED MESHES (비정렬 격자계에서 고차 정확도의 내재적 불연속 갤러킨 기법의 개발)

  • Lee, H.D.;Kwon, O.J.
    • Journal of computational fluids engineering
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    • v.12 no.3
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    • pp.29-40
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    • 2007
  • An implicit discontinuous Galerkin method for the two-dimensional Euler equations was developed on unstructured triangular meshes. The method can achieve high-order spatial accuracy by using hierachical basis functions based on Legendre polynomials. Numerical tests were conducted to estimate the convergence order of numerical solutions to the Ringleb flow and the supersonic vortex flow for which analytic solutions are available. Also, the flows around a 2-D circular cylinder and an NACA0012 airfoil were numerically simulated. The numerical results showed that the implicit discontinuous Galerkin methods couples with a high-order representation of curved solid boundaries can be an efficient method to obtain very accurate numerical solutions on unstructured meshes.

DEVELOPMENT OF IMPLICIT DISCONTINUOUS GALERKIN METHOD ON UNSTRUCTURED MESHES (비정렬 격자계에서 내재적 불연속 갤러킨 기법의 개발)

  • Lee, H.D.;Kwon, O.J.
    • 한국전산유체공학회:학술대회논문집
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    • 2007.04a
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    • pp.30-40
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    • 2007
  • The implicit discontinuous Galerkin method for the two-dimensional Euler equations was developed on unstructured triangular meshes, which can achieve higher-order accuracy by wing hierachical basis functions based on Legendre polynomials. Numerical tests were conducted to estimate the convergence order of numerical solutions to the Ringleb flow and the supersonic vortex flow for which analytic solutions are available. And, the flows around a circle and a NACA0012 airfoil was also numerically simulated. Numerical results show that the implicit discontinuous Galerkin methods with higher-order representation of curved solid boundaries can be an efficient higher-order method to obtain very accurate numerical solutions on unstructured meshes.

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