关于误差分析的问题

水胖娃儿

有些弹性力学的问题可以用有限元解和解析解对比来看误差，那有些没有解析解的问题如何进行误差分析啊。

tonnyw

That's why we have the so-called a-posetriori error estimation which is still a popular subject in the academic circle.

Let assume that the equation we solve is the right one which reflects the true reality and the unknown exact solution is uex. We are going to use the finite element solution uh to approximate uex.

The quantity we are interested is some kind of functional in terms of F(uex - uh) which is unknown. If we play with mathematics, we can get the following:
lower bound <= F(uex - uh) <=upper bound
If we can get the lower and upper bounds, we know how far we are away from the true error. Currently, people we can do get the bounds for certain problems such as linear elliptic problem or parabolic problem. However, there is no commercial software that can do this type of error estimation.

Another way to do the error estimation is based on superconvergence invented by Zienkiewicz. By postprocessing the finite element solution, we can have a solution which is closer to the unknown exact solution than the original finite element solution. As far as I know, most commercial FE softwares have this capability. But we cannot get bounds in this case.

水胖娃儿

谢谢哈，迫不及待想看又关于有限元误差分析的书了

tonnyw

Enjoy reading it!