1
$\begingroup$

Suppose I am solving a pde with a solution known with a finite-difference method. I can represent it as $A_hu_h=f$ for some approximating matrix $A_h$. And I define the discrete norm in which I will analyze the convergence to be $$||e||^2=h\sum e_i^2$$. I have the estimate for the error $$||e||\leq C ||\tau||$$ for $\tau$ local error if first order in space and $C$ a stability bound for $A^{-1}$. Then, I want to see which error gives my program to confirm the theory. I pick a point on the grid and watch what happens as a double the mesh. Assume I see the error gets decreased by 2, so I suspect it is a linear convergence at that point. Thus, I can see only rate of convergence at one point. How can I guarantee that the convergence is uniform "everywhere" on the grid? to confirm the theory results, should not I measure the error at each point and see I have a linear convergence in order to claim that the error goes to zero linearly in a discrete norm defined above? Because I could state results in a similar norms such as discrete $L^{\infty}$ or discrete $L^1$, however, what I measure by the computer is the same: difference between numerical solution and the function value at the point.

What confuses me is the fact that before I implement the method, I can state error estimates in a variety of norms, however, how do I relate these theoretical approximations to the error that I can actually measure, since this is just a value at the point on the grid and is independent from the way I do my theoretical analysis?

Edit: I should rephrase the question: There are a number of papers, where the convergence is measured for a particular point on the grid and the results are stated to confirm the theoretical estimates that are done in discrete $L^{\infty}$ or $L^{2}$ for the error vector. Is this sufficient to measure only at one point to obtain the confirmation and why the results presented only for the error at one point?

$\endgroup$
  • $\begingroup$ Is your mesh uniform? $\endgroup$ – Paul Jul 11 '12 at 17:29
  • $\begingroup$ Not in general. I would like to know both cases. $\endgroup$ – Kamil Jul 11 '12 at 17:44
  • $\begingroup$ What do you mean with $e_i$? In uniform grids, you can get down to what's called the discretization error. You can prove with a Taylor expansion, that it's in the order of $h^2$ if you use the standard 5-point stencil on the Laplace equation. $\endgroup$ – vanCompute Jul 13 '12 at 20:09
  • $\begingroup$ $e_i:=u(T,x_i)-U(T,x_i)$, that is the difference between the numerical solution and actual function at the point for a parabolic pde $u_t=u_{xx}$. I don't think I want to do Taylor there since I can use Lax equivalence theorem to estimate the order. $\endgroup$ – Kamil Jul 14 '12 at 13:38
3
+25
$\begingroup$

How can I guarantee that the convergence is uniform "everywhere" on the grid? to confirm the theory results, should not I measure the error at each point and see I have a linear convergence in order to claim that the error goes to zero linearly in a discrete norm defined above?

Yes. Do that.

There are a number of papers, where the convergence is measured for a particular point on the grid and the results are stated to confirm the theoretical estimates that are done in discrete L∞ or L2 for the error vector. Is this sufficient to measure only at one point to obtain the confirmation

No.

and why the results presented only for the error at one point?

There is no way we can tell you that without a link to the paper(s) you're referring to.

$\endgroup$
  • $\begingroup$ Well played, sir. Well played. $\endgroup$ – Geoff Oxberry Jul 14 '12 at 10:44
  • $\begingroup$ I am a student and I am learning. I don't get what you mean, because I see papers, where the results on the numerical method are published and I see that it is measured in one point on the grid only. $\endgroup$ – Kamil Jul 14 '12 at 13:35
  • $\begingroup$ @Medan it sounds like your question is really why do some papers measure numerical error at only one point rather than studying its norm? If so, I suggest rephrasing your question and providing a link to the paper(s) in question. What I was pointing out is that you answered your own question about what one should do already. $\endgroup$ – David Ketcheson Jul 14 '12 at 18:27
  • $\begingroup$ yes, David, probably this is my question. In finite differences a lot is based on the study of the norm and use of Lax theorem, however, how I can confirm results for different norms then? I edited the original question, I hope this clarifies what I am confused about. $\endgroup$ – Kamil Jul 14 '12 at 20:04
  • 1
    $\begingroup$ @Medan Seriously, evaluate the norm numerically. 99% of authors that do something less are either lazy or trying to hide something. With a little practice, you'll learn how easy it is to "cook" some comparisons that you see in papers and how discerning norms are (provided the solution is rich enough and that you use the "right" norm). $\endgroup$ – Jed Brown Jul 15 '12 at 1:25

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.