How do I apply the L norms?

werder85 · January 16, 2011, 03:37

I know this is a real newbie question.
I want to apply the L2-norm for a laplace equation. How should I proceed?
Should I simply calculate the difference between the values of the current and the last iteration, then the sum of the squares of these values and finally the square root?
Is this norm used to determine when to stop iterating?

Thanks a lot!

praveen · January 16, 2011, 04:01

Comparing difference between current and previous solutions can be misleading. You should instead see how well the discrete equations are satisfied. For example, if you are trying to solve

you can measure the convergence in the norm

$\frac{ \| L_h u_h - f_h \|}{\| f_h \|}$

The norms can be L2 norms.

werder85 · January 17, 2011, 05:26

Thanks, that's very interesting!
If I still want to compare the solutions from the latest two iterations, do I have to treat the domensions sepparately or not? Obviusly, by applying a L2-norm for a 3D domain will naturally provide higher values than an L2-norm for a 1D domain (because will be more points in the grid).

Further can you recommend me a good documentation on applying L-norms in CFD?

praveen · January 18, 2011, 09:25

If you divide by norm of rhs as I wrote, number of grid points does not matter.

werder85 · January 19, 2011, 02:31

I'm implementing the basic laplace equation, i.e. the RHS is zero... How should I proceed?
Thanks!

walli · January 19, 2011, 04:15

If the right hand side is zero, the problem is pretty boring, isn't it? No boundary conditions, no source term?

werder85 · January 19, 2011, 06:05

Ohh...now I understand, my mistake. The RHS also includes the boundary conditions (which are specified in my problem)...

RameshK · January 22, 2011, 02:14

Hi

You can use the following norm for stopping your iterations. It just calculates the relative error in the iterations

total_error=sqrt((\sun_{j=1}^N(\frac{u^(n+1) - u^(n)}{u^n})^2)_j)

which corresponds to all the points in the grid. You can compare this to an epsilon value and if the condition is met the iteration stops. n in the above formula represents the iteration count.

January 16, 2011, 03:37	How do I apply the L norms?	#1
werder85 New Member Join Date: Dec 2010 Posts: 14 Rep Power: 15	I know this is a real newbie question. I want to apply the L2-norm for a laplace equation. How should I proceed? Should I simply calculate the difference between the values of the current and the last iteration, then the sum of the squares of these values and finally the square root? Is this norm used to determine when to stop iterating? Thanks a lot!

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January 16, 2011, 04:01		#2
praveen Super Moderator Praveen. C Join Date: Mar 2009 Location: Bangalore Posts: 342 Blog Entries: 6 Rep Power: 18	Comparing difference between current and previous solutions can be misleading. You should instead see how well the discrete equations are satisfied. For example, if you are trying to solve you can measure the convergence in the norm $\frac{ \\| L_h u_h - f_h \\|}{\\| f_h \\|}$ The norms can be L2 norms.

January 17, 2011, 05:26		#3
werder85 New Member Join Date: Dec 2010 Posts: 14 Rep Power: 15	Thanks, that's very interesting! If I still want to compare the solutions from the latest two iterations, do I have to treat the domensions sepparately or not? Obviusly, by applying a L2-norm for a 3D domain will naturally provide higher values than an L2-norm for a 1D domain (because will be more points in the grid). Further can you recommend me a good documentation on applying L-norms in CFD?

January 18, 2011, 09:25		#4
praveen Super Moderator Praveen. C Join Date: Mar 2009 Location: Bangalore Posts: 342 Blog Entries: 6 Rep Power: 18	If you divide by norm of rhs as I wrote, number of grid points does not matter.

January 19, 2011, 02:31		#5
werder85 New Member Join Date: Dec 2010 Posts: 14 Rep Power: 15	I'm implementing the basic laplace equation, i.e. the RHS is zero... How should I proceed? Thanks!

January 19, 2011, 04:15		#6
walli New Member Join Date: May 2009 Posts: 28 Rep Power: 16	If the right hand side is zero, the problem is pretty boring, isn't it? No boundary conditions, no source term?

January 19, 2011, 06:05		#7
werder85 New Member Join Date: Dec 2010 Posts: 14 Rep Power: 15	Ohh...now I understand, my mistake. The RHS also includes the boundary conditions (which are specified in my problem)...

January 22, 2011, 02:14	Hi	#8
RameshK Member Ramesh K Join Date: Dec 2009 Location: Bangalore Posts: 73 Rep Power: 16	Hi You can use the following norm for stopping your iterations. It just calculates the relative error in the iterations total_error=sqrt((\sun_{j=1}^N(\frac{u^(n+1) - u^(n)}{u^n})^2)_j) which corresponds to all the points in the grid. You can compare this to an epsilon value and if the condition is met the iteration stops. n in the above formula represents the iteration count.