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I am looking at the pde of the type $u_t=\mathcal{L}u$ for some elliptic operator $\mathcal{L}$ and on some domain $D$. Assume I am solving that with a finite difference method and want to estimate the error. Thus, I create a mesh over $D$ and compute the error in discrete $L^2$ norm, that is at every point on the grid I take the difference between the function itself and an approximate solution, square it and sum up. This sum includes all the points, including the boundary points. By Lax theorem the rate of convergence is the rate of local truncation error PROVIDED the problem is well posed and the derivatives in local truncation error are bounded. However, assume it can be shown that the weak solution exist and unique and is sufficiently smooth inside of the domain but on the boundaries it is only satisfied in a weak sense, and lacks regularity. So there is no hope that the terms in local truncation error are bounded on the boundary, they are not even defined there. Thus, how do all these points on the boundary contribute to the error? They exist even if we refine a mesh, more than that we add points with it. Please let me know should I take into account irregularity on the boundary.

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    $\begingroup$ It seems difficult to answer this question because in my opinion it is not precisely defined. I suggest that you use mathematics where possible (rather than just words). Also, your statement of the Lax equivalence theorem is not correct. $\endgroup$ – David Ketcheson Jul 15 '12 at 20:05
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Elliptic problems have weak singularities at reentrant corners, therefore you must interpret solutions weakly, making strong form discretizations like finite differences less natural than weak form discretizations (like finite element methods) to analyze and compute with. The canonical analysis text is

  • Grisvard (1985) Elliptic problems in nonsmooth domains.

Solution of elliptic systems in domains with reentrant corners is the canonical problem in adaptive finite element methods, thus the topic of thousands of papers. A good place to get started on the background theory is

  • Brenner and Scott (1994,2002,2008) The mathematical theory of finite element methods.

There are numerous open source software packages that use adaptive refinement guided by a posteriori error estimators to solve elliptic problems in domains with reentrant corners.

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This is one of the many reasons why the analysis of finite element methods is so much simpler than that of finite difference methods...

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    $\begingroup$ not sure that answers my question... $\endgroup$ – Kamil Jul 8 '12 at 23:58
  • $\begingroup$ No, I meant to say that the analysis of the FEM is so much simpler because it uses integrals of functions, not point values. For problems with singularities of the kind that you can get with $H^1$ functions, the integrals are all well defined and so the analysis is simple. But point values don't always exist, and consequently the analysis of the finite difference method is difficult. $\endgroup$ – Wolfgang Bangerth Jul 9 '12 at 13:42
  • $\begingroup$ @WolfgangBangerth: You wrote "finite element methods ... finite element methods." We suspect that one of those should be "finite difference methods," and I would suggest that it should be the latter one... $\endgroup$ – Bill Barth Jul 15 '12 at 16:20
  • $\begingroup$ Ah yes. Thanks for pointing it out, I've edited it. $\endgroup$ – Wolfgang Bangerth Jul 15 '12 at 18:29

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