Sobolev方程的半有限元方法

The half finite element method for Sobolev equation

对Sobolev方程采用半有限元法进行数值模拟.通过将空间变量和时间变量分离,得到Sobolev方程的离散格式.首先对空间变量应用有限元方法进行离散化,得到常微分方程组的初值问题;再对时间变量应用有限差分法进行离散化,得到一系列线性方程组,求解可得到Sobolev方程的数值解.本文从理论上推导出了本文所讨论的Sobolev方程半有限元算法的矩阵算法格式,分析了其可行性.在最后给出了数值例子,从数值例子中进一步验证了半有限元方法的可行性.

In this paper, a half finite element method was proposed to solve the Sobolev equation. Discretization of Sobolev equation was presented by separating space and time variables. Firstly, applying finite element method to get discretization of the spatial var iables, the problem was transformed into the initial value problems of ordinary differen- tial equations. Secondly, applying finite-difference method to get discretization of the time variables, the problem was transformed into a series of linear equations. Thirdly, Numerical solution of the Sobolev equation was acquired by solving the series of linear e- quations. The method was proved to be feasible by theoretical analysis the Matrix algo- rithm format of the Sobolev equation was presented. Finally, numerical results were provided to illustrate the efficiency of our method.