Several Finite Volume Element Schemes And Some Application In Radiation Heat Conduction Problems

Since the Finite Volume Method can preserve local conservation of cer-tain physical quantities, such as mass, energy and so on, it has become one of the important numerical methods in scientific and engineering computa-tion. The Finite Volume Element (FVE) Method is an important kind of the Finite Volume Methods. Although there existed many literatures about the FVE Method, there still have many problems to study. The Inertial Con-finement Fusion (ICF) problem is one type of fluid mechanic problems about fusional plasma. The efficient numerical methods for the two dimensional three temperatures (2D3T) radiation heat conduction equations are crucial for stimulating ICF problem, because it dominates the CPU time during nu-merical solving. Hence, it is a meaningful work to construct efficient FVE scheme and to design fast linear solver for the radiative heat conduction problem. Serval FVE schemes are systemically discussed on the stationary diffusion problem and radiative heat conduction problem in this dissertation, respectively.The pointwise asymptotic expansions of the isoparametric bilinear finite volume element is firstly derived and proved on uniform grids for the station-ary diffusion problem with smooth variant coefficients. Two corresponding finite volume element schemes are proposed on the quadrilateral grids with mixed cells. Numerical results confirm the theoretical results, and show the saturated convergent order of the second new scheme on mixed grids and the superconvergence in the sense of pointwise for parabolic problem.Firstly, two preserving-symmetry finite volume element (SFVE) schemes for the stationary diffusion problem are established on unstructure quadrilat-eral grids. The saturated order of error in both L2-norm and H1-norm for the discrete solutions under quasi-uniform partition is proved when the diffusion coefficient is smooth. Secondly, two SFVE schemes are similarly obtained on the mixed-quadrilateral grids. Finally, one SFVE scheme is designed for the two dimensional elliptic problem with nonlocal boundary conditions, and the optimal error estimates in the L2 norm are derived. Numerical experiments verify the theoretical results and show similar traits about the schemes on mixed grids.Based on the hierarchical basis, the spectral equivalence is firstly estab-lished for two kinds of stiff matrices from the quadratic finite element and second order mixed-type finite volume element method, then a new precondi-tioner for GMRES is proposed. Two second-order mixed-type finite volume element schemes are constructed on mixed quadrilateral grids. Numerical results confirm our theoretic analysis, show the efficiency and robustness of our preconditioner, and show similar traits about the schemes on mixed grids as above.A SFVE scheme and a second order mixed-type finite volume element scheme are proposed for the two dimensional radiative heat conduction prob-lem about cylindrical transport pipe. The energy conservative error is an-alyzed for the SFVE scheme. Numerical experiments show that the energy conservative error of new scheme is small, and numerical simulation results are consistent with the actual physical phenomena.The new schemes on mixed grids above for the stationary diffusion prob-lem are applied to solving radiative heat conduction problems about 2D3T. Numerical results show that the energy conservative errors of new schemes are small and acceptable, and the heat conduction process is consistent with the actual physical phenomena. All numerical results indicte that our new FVE schemes are effective and robust.