【作者】 张胜；

【导师】 李立康；

【作者基本信息】 复旦大学 ， 计算数学， 2003， 博士

【摘要】 近年来，有限体积法因其具有局部守衡的性质且实施起来相对较为简单灵活而被广泛地应用于求解许多数学物理问题。本文用有限体积法离散多边形区域(可能非凸)上的二阶非对称不定椭圆问题。目前关于有限体积法的研究论文越来越多，但有限体积法的理论还很不完善。并且大部分已有研究结果均集中于给出对某一具体问题用有限体积法离散后的误差估计，而对于如何高效求解其离散方程，这一无论从理论上讲，还是从实际应用角度出发都具有重要意义和巨大实用价值的问题，目前这方面的研究结果还很少。本文研究了有限体积法对多边形区域(可能非凸)上的二阶非对称不定椭圆问题进行离散后所得离散方程的高效求解问题，分别构造了求解其离散方程的层次基方法、区域分解方法和预条件的GMRES方法，并证明了其收敛性。更多还原

【Abstract】 It’s well known that the covolume methods (finite volume el-ement methods) are widely and successfully used in solving many mathematical-physical problems due to their simplicity and local conservation properties in recent years. In this paper, we use covol-ume method to discrete second order norisyrnmetric and indefinite elliptic problem on a polygonal domain (possibly nonconvex). For solving the corresponding discretization equation, there are few re-sults on the construction of efficient solvers. Most existing results only presented the related error estimate for a concrete problem discretized by covolume methods. In this paper, hierarchical basis method, domain decomposition method and precondtioned GMRES method are constructed. The convergence of these method are also verified.更多还原