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The hybrid difference methods for convection-diffusion equations and interface problems
- Author(s)
- TRAN MAI LAN
- Advisor
- Youngmok Jeon
- Department
- 일반대학원 수학과
- Publisher
- The Graduate School, Ajou University
- Publication Year
- 2018-08
- Language
- eng
- Alternative Abstract
- In this study we aim to introduce and analyze the hybrid difference methods (HDMs) for convection-diffusion equations and the elliptic interface problems. The HDMs are composed of the finite difference approximation of the PDEs within cells (cell FD) and the finite difference approximation of the continuity of normal fluxes on the edges of cells (intercell FD). Numerical analysis of the HDMs is possible in a variational setting as other hybridized finite element methods. Moreover, the HDMs significantly reduce degrees of freedom and work well on non-uniform grids retaining convergence order. For convection dominated diffusion equations we propose two methods so-called the upwind hybrid difference method and the penalized upwind method. It is well known that the solutions of convection-diffusion equations typically contain interior and boundary layers and numerical solutions obtained by standard methods usually exhibit spurious and nonphysical oscillations. The upwind hybrid difference method is stable but numerical solution may contain oscillations due to the small diffusion term. By adding a penalty parameter in the intercell finite difference, which is oftenly chosen in the form of some power of the grid size, we obtain the penalized upwind method. Numerical experiments show that with an appropriate choice of the penalty parameter the penalized upwind method can give access to controlling oscillation as well as smearing of the numerical solutions. The hybrid difference approach can also be used to solve the elliptic interface problems. For that we introduce the interface hybrid difference (IHD) method. The domain is decomposed into cells so that each cell is contained exclusively in one of the subregions of the domain. The mesh does not required to be uniform and it is constructed to satisfy the deviation condition. The cells away from the interface are regular rectangles and the cells adjacent to the interface are trapezoids. In two dimensional case, a new intercell condition is introduced when the cell boundary coincides with the interface. If the interface is a polygon then the interface hybrid difference method is proved to have an optimal order of convergence in the energy norm. Several numerical experiments are presented to confirm the theoretical results.
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- Graduate School of Ajou University > Department of Mathematics > 3. Theses(Master)
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