Relation between Solution Error Behavior and Polynomial Approximation Degree

[JAEKWANG KIM]

Hi all, I have question on error behavior of FEM. 

I thought that the order of error is O(h^p) where h is a mesh-size and p is polynomial degree we use in approximation. 

So, I thought that if I plot an error with number of mesh in log-log scale, than the graph will show -p slope. 

However, I the error behaves little bit different from my expectation.

For example, I use a step7 tutorial program (which solves Helmholtz decomposition and compares the FEM solution with exact solution.) 

The error curve showed more steep slope whenever I increase polynomial degree approximation however, the slope is not (-p). 

I reached slope (-3) when I used fifth-degree polynomial approximation...   

You can check this behavior in attached picture. 

Until now, I have considered, 

1. Mapping(From reference cell to real cell) degree (which is originally set to 1 but I used higher mapping) 

2. Instead of Qgauss quadrature, I am using QgaussLobatto Quadrature for any integration over cells. 

3. Shape function , again I tried to use QgaussLobatto node point for this....) 

is there any suggestion that I need to fix more? 

or my first prediction that the slope will show '-p' or error will just behave O(h^p) was wrong?

I am always thank you for all guys!

Jaekwang Kim  

[Wolfgang Bangerth]

It ought to be O(h^p), but I *think* what you are plotting is O(N^{-s})
where N is the number of degrees of freedom. Since N=O(h^{-2}) in 2d,
what you see is a convergence order of p=2s. Does that explain the problem?

Best
W.


--
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Wolfgang Bangerth email: bang...@colostate.edu
www: http://www.math.colostate.edu/~bangerth/

[JAEKWANG KIM]

Yes! your explanation clear explains the situation ! 

Thank you!!!

Jaekwang Kim 

2016년 9월 29일 목요일 오전 11시 25분 28초 UTC-5, Wolfgang Bangerth 님의 말:

[seven]

Hello Jaekwang,

I am trying to generate some log-log plots, and wondering if you used the functions in deal.ii to generate the figure. If not, what did you use?

Thanks,

Jiaqi

在 2016年9月29日星期四 UTC-4上午11:41:48，Jaekwang Kim写道：

[Jaekwang Kim]

No, I typically, calculate values itself and use MATLAB to draw plots. 

Thanks,

Jaekwang 

[seven]

I see. 

Thanks,

Jiaqi

在 2017年11月27日星期一 UTC-5下午6:31:54，Jaekwang Kim写道：