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The upwind hybrid difference methods for a convection diffusion equation
Jeon, Youngmok,Tran, Mai Lan Elsevier 2018 Applied numerical mathematics Vol.133 No.-
<P><B>Abstract</B></P> <P>We propose the upwind hybrid difference method and its penalized version for the convection dominated diffusion equation. The hybrid difference method is composed of two types of approximations: one is the finite difference approximation of PDEs within cells <I>(cell FD)</I> and the other is the <I>interface finite difference (interface FD)</I> on edges of cells. The interface finite difference is derived from continuity of normal fluxes. The penalty method is obtained by adding small diffusion in the interface FD. The penalty term makes it possible to reduce severe numerical oscillations in the upwind hybrid difference solutions. The penalty parameter is designed to be some power of the grid size. A complete stability is provided. Convergence estimates seems to be conservative according to our numerical experiments. To exposit convergence property and controllability of numerical oscillations several numerical tests are provided.</P>
A CELL BOUNDARY ELEMENT METHOD FOR A FLUX CONTROL PROBLEM
Jeon, Youngmok,Lee, Hyung-Chun Korean Mathematical Society 2013 대한수학회지 Vol.50 No.1
We consider a distributed optimal flux control problem: finding the potential of which gradient approximates the target vector field under an elliptic constraint. Introducing the Lagrange multiplier and a change of variables the Euler-Lagrange equation turns into a coupled equation of an elliptic equation and a reaction diffusion equation. The change of variables reduces iteration steps dramatically when the Gauss-Seidel iteration is considered as a solution method. For the elliptic equation solver we consider the Cell Boundary Element (CBE) method, which is the finite element type flux preserving methods.
Elsevier 2010 Journal of computational and applied mathematics Vol.234 No.8
<P><B>Abstract</B></P><P>We introduce two kinds of the cell boundary element (CBE) methods for convection dominated convection–diffusion equations: one is the CBE method with the exact bubble function and the other with inexact bubble functions. The main focus of this paper is on inexact bubble CBE methods. For inexact bubble CBE methods we introduce a family of numerical methods depending on two parameters, one for control of interior layers and the other for outflow boundary layers. Stability and convergence analysis are provided and numerical tests for inexact bubble CBEs with various choices of parameters are presented.</P>