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A note on necessary and sufficient conditions for convergence of the finite element method. (English) Zbl 1363.65189
Brandts, J. (ed.) et al., Proceedings of the international conference ‘Applications of mathematics’, Prague, Czech Republic, November 18–21, 2015. In honor of the birthday anniversaries of Ivo Babuška (90), Milan Práger (85), and Emil Vitásek (85). Prague: Czech Academy of Sciences, Institute of Mathematics (ISBN 978-80-85823-65-3). 132-139 (2015).
The author presents some geometrical results concerning the convergence of the finite element method. While there is plenty of literature on the sufficient condition for the optimal \(O(h)\) convergence of the finite element method such as the maximum angle condition [I. Babuška and A. K. Aziz, SIAM J. Numer. Anal. 13, 214–226 (1976; Zbl 0324.65046)], few results are known for the necessary condition.
Recently, A. Hannukainen et al. [Numer. Math. 120, No. 1, 79–88 (2012; Zbl 1255.65196)] pointed out that the maximum angle condition is not always necessary for \(O(h)\) convergence. They gave an example of mesh division which violates the maximum angle condition but realizes \(O(h)\) convergence. Their example is constructed in such a way that they subdivide the original triangular mesh which satisfies maximum angle condition. In their example, \(O(h)\) convergence is easily shown by Céa’s lemma.
The author noticed that their example consists of two kinds of triangles, triangles which satisfy the maximum angle condition and triangles which violate the maximum angle condition. Furthermore, the author proved that by mesh subdivision, one cannot construct triangular mesh containing only degenerate triangles. Based on these facts, the author presented a hypothesis on a necessary condition for \(O(h)\) convergence. That is, “In order to realize \(O(h)\) convergence, triangular elements satisfying the maximum angle condition must be dense”.
Though the pressent author was unable to prove that, I feel that this hypothesis is reasonable and true. However, if this hypothesis has proved, I think that there still remains the wide gap between necessary condition and sufficient condition. Anyway, since no other trivial result for necessary condition is known, the content of this paper is quite meaningful.
For the entire collection see [Zbl 1329.00187].

MSC:
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
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