【作者】 李琳琳；

【导师】 李德茂；

【作者基本信息】 内蒙古大学 ， 计算数学， 2006， 硕士

【摘要】 具有奇异系数的抛物方程是近年来在核物理、气体动力学、流体力学、边界层理论、非线性场和光学等实际问题中提出的一类重要方程，数值分析和求解该类方程具有重要意义。而有限元方法现在已经成为解决科学技术各个领域里出现的偏微分方程的求解近似解的一种有力方法。计算数学工作者们利用有限差分、对称有限元、非对称有限元等方法对具有奇异系数的椭圆、抛物方程进行了深入的研究，得到了一系列很好的结果。 本文考虑一类四阶非线性奇异抛物方程的有限元方法。首先证明了相应变分问题弱解的存在唯一性；其次给出了半离散解的加权L₂模及加权H²模误差估计；然后又给出了全离散解即C—N方法的加权L₂模误差估计；最后又考虑了四阶拟线性奇异抛物方程的半离散解的后验误差估计。更多还原

【Abstract】 The parabolic equations with singular coeffoients are very important ones in practical problems. Such as in nuclear physics, gas dynamics, mochanics, boundary layer theory, nonlinear filed and optics. The finite element method （FEM） is now a classical method for solving the partial differential equations arising in various contexts of science and technology. In recent years, many researchers have studied the partial differential equations with singular coefficients by finite difference method, symmetric finite element method, nonsymmetric finite element method and so on. Some ideal results have been obtained.In this article, a class of fourth order nonlinear boundary problems with singular coefficient is considered. First of all, existence and uniqueness of the weak solution of the variational problem is proved; Secondly, the weighted L₂ norm, the weighted H² norm error estimation for semi-discrete problems were obtained; Thind, error estimation in weighted L₂ norm for fully-discrete problems was obtained; At last , we give the posteriori error estimation with finite element semi-discrete methods for another nonlinear singular parabolic problem.更多还原