【作者】 刘金存；

【导师】 李德茂； 李宏；

【作者基本信息】 内蒙古大学 ， 计算数学， 2005， 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,许多专家学者利用差分法、投影法等方法对此类问题进行了深入研究,得到了很好的结果,利用有限元方法对奇异微分方程的研究,目前主要集中在传统的有限元方法,而近几年,有限元方法迅速发展,出现了时空有限元方法、运动有限元方法、间断有限元方法等各种非标准有限元方法,利用非标准有限元方法对奇异微分方程进行研究的文章至今还未见到。 本文利用时间间断空间连续的间断时空非标准有限元方法,对奇异线性、半线性抛物方程进行了理论误差分析。对线性问题,在对时空网格施加一定的限制条件下,利用对偶问题的稳定性和Galerkin正交性给出了在时间结点处的超收敛估计结果和在时间区间内部的最佳阶估计结果;而对于半线性抛物方程,则利用有限元方法和有限差分方法相结合的技巧,不对时空网格施加限制条件,证明了弱解的存在唯一性,给出了时间最大模,空间加权L₂模误差估计。更多还原

【Abstract】 The parabolic equations with singular coefficients are very important ones in practical problems e.g. problems in nuclear physics, gas dynamics, mechanics, boundary layer theory, nonlinear field and optics. In recent years, many researchers have studied such kind of problems by difference method, projection method, etc.. And they have achieved excellent results. As to the finite element methods （FEM） for singular differential equations, the main achievements were gained by traditional FEM. In recent years, the FEM developed quickly. Many nonstandard finite element methods were proposed, such as space-time FEM, moving FEM and discontinuous FEM, etc.. However, the papers which are on the discussion of singular differential equations solved by nonstandard FEM are not appeared as so for.In this paper, we give the theoretical error estimates on linear and semi-linear singular parabolic equations by discontinuous space-time nonstandard FEM which is discontinuous in time but continuous in space. For linear problem, we give the superconvergent results at the nodes of time and the optimal order estimate in the interior of the time intervals, respectively. These results are gained through the stability of dual problems and Galerkin orthogonality with the limitation of the space-time meshes; For the semi- linear equation, the technique of combination of the FEM and finite difference method is adopted to prove the existence and uniqueness of the weak solution without the limitation of the space-time meshes. And the error estimate for maximum norm in time, weighted L₂ -norm in space is obtained.更多还原