• Bulletin of the Korean Mathematical Society
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• v.50 no.3
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• pp.963-982
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• 2013

The Galerkin method has been developed mainly for 2nd order differential equations. To get numerical solutions, there are some choices of Riesz bases for the approximation subspace $V_j{\subset}L^2$. In this paper we shall propose a uniform approach to find suitable Riesz bases for higher order differential equations. Especially for the beam equation (4-th order equation), we also report numerical results.

• 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.

• Journal of Korean Society of Coastal and Ocean Engineers
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• v.30 no.4
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• pp.180-190
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• 2018

The preliminary study was carried out to utilize the rolling porous pendulum plate as a hybrid system combining blocking and extracting of wave energy. The Galerkin method suggested by Porter and Evans (1995) was used to solve the diffraction and radiation problems to obtain reflection and transmission coefficient, roll displacement, extracted power. The Galerkin method provides better convergence than the matched eigenfunction expansion method (MEEM), which improves the accuracy of the analytical solution even if the CPU time is shorter. The porous plate can not be said to be more effective than the impermeable plate in terms of wave energy extraction and wave blocking, but it has the advantage of reducing the wave load and exchanging seawater.

• Journal of the Chungcheong Mathematical Society
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• v.10 no.1
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• pp.57-70
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• 1997

In this paper two preconditioned GMRES and QMR methods are applied to the non-Hermitian system from the Petrov-Galerkin procedure for the Poisson equation and compared to each other. To our purpose the ILUT and the SSOR preconditioners are used.

• 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.

• 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.

• 한국전산유체공학회:학술대회논문집
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• 2008.03a
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• pp.57-70
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• 2008

A high-order accurate Euler flow solver based on a discontinuous Galerkin finite-element method has been developed for the numerical simulations of blade-vortex interaction phenomena on unstructured meshes. A free vortex in freestream was investigated to assess the vortex-preserving property and the accuracy of the present flow solver. Blade-vortex interaction problems in subsonic and transonic freestreams were simulated by adopting a multi-level solution-adaptive dynamic mesh refinement/coarsening technique. The results were compared with those of other numerical and experimental methods. It was shown that the present discontinuous Galerkin flow solver can preserve the vortex structure for significantly longer vortex convection time and can accurately capture the complex unsteady blade-vortex interaction flows, including generation and propagation of acoustic waves.

• 한국전산유체공학회:학술대회논문집
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• 2008.10a
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• pp.57-70
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• 2008

A high-order accurate Euler flow solver based on a discontinuous Galerkin finite-element method has been developed for the numerical simulations of blade-vortex interaction phenomena on unstructured meshes. A free vortex in freestream was investigated to assess the vortex-preserving property and the accuracy of the present flow solver. Blade-vortex interaction problems in subsonic and transonic freestreams were simulated by adopting a multi-level solution-adaptive dynamic mesh refinement/coarsening technique. The results were compared with those of other numerical and experimental methods. It was shown that the present discontinuous Galerkin flow solver can preserve the vortex structure for significantly longer vortex convection time and can accurately capture the complex unsteady blade-vortex interaction flows, including generation and propagation of acoustic waves.

• 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.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.13 no.3
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• pp.169-180
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• 2009

A discontinuous Galerkin method with interior penalty terms is presented for linear Sobolev equation. On appropriate finite element spaces, we apply a symmetric interior penalty Galerkin method to formulate semidiscrete approximate solutions. To deal with a damping term $\nabla{\cdot}({\nabla}u_t)$ included in Sobolev equations, which is the distinct character compared to parabolic differential equations, we choose special test functions. A priori error estimate for the semidiscrete time scheme is analyzed and an optimal $L^\infty(L^2)$ error estimation is derived.