• 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.

• 한국전산구조공학회:학술대회논문집
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• 한국전산구조공학회 1998년도 봄 학술발표회 논문집
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• pp.68-74
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• 1998

A finite element analysis for physical phenomenon which are governed by parabolic equation, has some inefficiencies caused by much computational time and large storage space. In this paper, a comparative study is performed to suggest the best efficient transient analysis algorithms for parabolic equations. First, the general finite element analysis techniques are summarized in views of formulation procedures, treatments of convection terms. and time stepping methods. Results of several combinations applied to one dimensional convection-diffusion equation and Burger equation are represented and compared using some criteria such as accuracy, stability, and computational time. Through the results, some guidelines to select a algorithm for solving parabolic equations are proposed for diffusion dominant and convection dominant cases. Finally applicability of two dimensional extension of the result is also discussed.

• 호남수학학술지
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• 제37권3호
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• pp.299-315
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• 2015

In this paper, we study the first-order system least-squares (FOSLS) spectral method for parabolic partial differential equations. There were lots of least-squares approaches to solve elliptic partial differential equations using finite element approximation. Also, some approaches using spectral methods have been studied in recent. In order to solve the parabolic partial differential equations in parallel, we consider a parallel numerical method based on a hybrid method of the frequency-domain method and first-order system least-squares method. First, we transform the parabolic problem in the space-time domain to the elliptic problems in the space-frequency domain. Second, we solve each elliptic problem in parallel for some frequencies using the first-order system least-squares method. And then we take the discrete inverse Fourier transforms in order to obtain the approximate solution in the space-time domain. We will introduce such a hybrid method and then present a numerical experiment.

• 한국수학사학회지
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• 제24권1호
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• pp.31-44
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• 2011

라이만 다양체 위에서의 편미분 방정식의 연구는 미분기하학에서 중요한 연구 분야로 인식되어 왔다. 본 논문에서는 특히 최근에 미분기하학과 위상수학 분야에서 중요한 역할을 하고 있는 리이만 다양체 위에서의 포물형 방정식에 관한 역사적으로 주목받고 있는 중요한 연구 결과를 정리해 보고, 아울러 이 분야의 최근 연구 결과를 고찰한다.

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제15권1호
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• pp.19-34
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• 2008

A new algorithm for the solution of an inverse problem of determining unknown source parameter in a parabolic equation in reproducing kernel space is considered. Numerical experiments are presented to demonstrate the accuracy and the efficiency of the proposed algorithm.

• 대한수학회보
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• 제45권1호
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• pp.169-190
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• 2008

Second-order parabolic equations with variable coefficients are considered on $\mathbb{R}^d$ and $C^1$ domains. Existence and uniqueness results are given in $L_q(L_p)$-spaces, where it is allowed for the powers of summability with respect to space and time variables to be different.

• 대한수학회보
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• 제53권4호
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• pp.1123-1148
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• 2016

The aim of this note is to provide detailed proofs for local estimates near the boundary for weak solutions of second order parabolic equations in divergence form with time-dependent measurable coefficients subject to Neumann boundary condition. The corresponding parabolic equations with Dirichlet boundary condition are also considered.

• 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.

• Kyungpook Mathematical Journal
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• 제60권1호
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• pp.145-161
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• 2020

In this article, we study the solvability of the Cauchy-Dirichlet problem for a class of nonlinear parabolic equations with nonstandard growth and nonlocal terms. We prove the existence of weak solutions of the considered problem under more general conditions. In addition, we investigate the behavior of the solution when the problem is homogeneous.

• Journal of applied mathematics & informatics
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• 제9권2호
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• pp.635-643
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• 2002

The asymptotic behaviour of the solutions of initial - boundary value problem for a class of nonlinear parabolic differential - functional equations is studied via the method of differential inequalities in order to obtain oscillation criterion for the solutions.