Question about grid independence study

Anna Tian · October 24, 2012, 11:18

Hi,

Why we need 3 points on the parabolic grid independence study curve? Why not only 2? Why all of the 3 points have to be on the parabolic curve? In the case that my medium mesh simulation result is very close to the fine mesh but my coarse mesh is totally out of the parabolic curve, why I still can't say my fine mesh is good enough to use?

Thank you very much!

Jonas Holdeman · October 25, 2012, 10:04

If I understand your question correctly, the convergence rate upon mesh refinement, assumed parabolic in your case, is an assymptotic relation, valid only for sufficiently refined meshes. Three points on a parabolic curve doesn't prove your mesh is fine enough for the relation to apply, but may give you some confidence that it is. Two points don't tell you anything about assymptotic convergence rate.

Anna Tian · October 25, 2012, 11:15

Why three points on parabolic curve still can't prove my mesh is good enough? What can prove that?

Quote:

Originally Posted by Jonas Holdeman

If I understand your question correctly, the convergence rate upon mesh refinement, assumed parabolic in your case, is an assymptotic relation, valid only for sufficiently refined meshes. Three points on a parabolic curve doesn't prove your mesh is fine enough for the relation to apply, but may give you some confidence that it is. Two points don't tell you anything about assymptotic convergence rate.

Jonas Holdeman · October 25, 2012, 15:02

Suppose the convergence was actually linear, given asymptotically by a0*h+a1*h^2+a2*h^3+...., but where a0 was very small. Then there might be a range of h where a0*h << a1*h^2. Then in this range the assymptptic convergence would appear to be quadratic. But if you made h small enough, the linear term would begin to dominate. So you can't prove by giving examples that the convergence is quadratic, but if points continue to lie on the quadratic curve as you decrease h, you might gain confidence that quadratic convergence is indeed the case.

FMDenaro · October 25, 2012, 15:13

See Section 3.9 in the book of Ferziger and Peric, three grids give you an estimation of the rate of convergence

October 24, 2012, 11:18	Question about grid independence study	#1
Anna Tian Senior Member Meimei Wang Join Date: Jul 2012 Posts: 494 Rep Power: 16	Hi, Why we need 3 points on the parabolic grid independence study curve? Why not only 2? Why all of the 3 points have to be on the parabolic curve? In the case that my medium mesh simulation result is very close to the fine mesh but my coarse mesh is totally out of the parabolic curve, why I still can't say my fine mesh is good enough to use? Thank you very much! __________________ Best regards, Meimei

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October 25, 2012, 10:04		#2
Jonas Holdeman Senior Member Jonas T. Holdeman, Jr. Join Date: Mar 2009 Location: Knoxville, Tennessee Posts: 128 Rep Power: 18	If I understand your question correctly, the convergence rate upon mesh refinement, assumed parabolic in your case, is an assymptotic relation, valid only for sufficiently refined meshes. Three points on a parabolic curve doesn't prove your mesh is fine enough for the relation to apply, but may give you some confidence that it is. Two points don't tell you anything about assymptotic convergence rate.

October 25, 2012, 15:02		#4
Jonas Holdeman Senior Member Jonas T. Holdeman, Jr. Join Date: Mar 2009 Location: Knoxville, Tennessee Posts: 128 Rep Power: 18	Suppose the convergence was actually linear, given asymptotically by a0h+a1h^2+a2h^3+...., but where a0 was very small. Then there might be a range of h where a0h << a1*h^2. Then in this range the assymptptic convergence would appear to be quadratic. But if you made h small enough, the linear term would begin to dominate. So you can't prove by giving examples that the convergence is quadratic, but if points continue to lie on the quadratic curve as you decrease h, you might gain confidence that quadratic convergence is indeed the case.

October 25, 2012, 15:13		#5
FMDenaro Senior Member Filippo Maria Denaro Join Date: Jul 2010 Posts: 6,849 Rep Power: 73	See Section 3.9 in the book of Ferziger and Peric, three grids give you an estimation of the rate of convergence