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On the interpolation constants over triangular elements. (English) Zbl 1363.65014

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). 110-124 (2015).
Let \(T \subset \mathbb{R}^2\) be a triangle with the vertices \(p_i\) and the edges \(\gamma_i\) (\(i=1,2,3\)). Let \(H^k(T)\) (\(k=1,2\)) be the usual Sobolev spaces defined on \(T\). Define the functions spaces as follows: \[ \begin{aligned} V^{1,1}(T) & = \left\{\varphi \in H^1(T) \Bigm| \int_T \varphi dx dy = 0\right\}, \\ V^{1,2}(T) & = \left\{\varphi \in H^1(T) \Bigm| \int_{\gamma_i} \varphi ds = 0, \; i=1,2,3\right\}, \\ V^{2}(T) & = \left\{\varphi \in H^2(T) \Bigm| \varphi(p_i) = p_i, \; i=1,2,3\right\}.\end{aligned} \] Then, the author tried to obtain upper bound of the following quantities: \[ \begin{aligned} C_1(T) = \sup_{u \in V^{1,1}(T)} \frac{\|u\|_{L^2(T)}}{\|\nabla u\|_{L^2(T)}}, & \qquad C_2(T) = \sup_{u \in V^{1,2}(T)} \frac{\|u\|_{L^2(T)}}{\|\nabla u\|_{L^2(T)}}, \\ C_3(T) = \sup_{u \in V^{2}(T)} \frac{\|u\|_{L^2(T)}}{|u|_{H^2(T)}}, & \qquad C_4(T) = \sup_{u \in V^{2}(T)} \frac{\|\nabla u\|_{L^2(T)}}{|u|_{H^2(T)}}.\end{aligned} \] The results of this paper are as follows: \[ \begin{aligned} C_1(T) < K_1(T) & = \sqrt{ \frac{A^2 + B^2 + C^2}{28} - \frac{S^4}{A^2B^2C^2}}, \\ C_2(T) < K_2(T) & = \sqrt{ \frac{A^2 + B^2 + C^2}{54} - \frac{S^4}{2A^2B^2C^2}}, \\ C_3(T) < K_3(T) & = \sqrt{ \frac{A^2B^2 + B^2C^2 + C^2A^2}{83} - \frac{1}{24} \left(\frac{A^2B^2C^2}{A^2+B^2+C^2}+ S^2\right)}, \\ C_4(T) < K_4(T) & = \sqrt{ \frac{A^2B^2C^2}{16S^2} - \frac{A^2 + B^2 + C^2}{30} -\frac{S^2}{5}\left(\frac{1}{A^2} + \frac{1}{B^2} + \frac{1}{C^2} \right)},\end{aligned} \] where \(A\), \(B\), \(C\) are the edge lengths, and \(S\) is the area of \(T\), respectively.
These estimations are a bit surprising. In particular, the fourth estimation \(C_4(T) < K_4(T)\) is very important for error analysis of finite element method by the following reason.
Let \(v \in H^2(T)\) be given. Then, its Lagrange interpolation \(\Pi_h v\) is a polynomial of degree one defined by \(\Pi_h v(p_i) = v(p_i)\), \(i=1,2,3\). By the definition, we have \(v - \Pi_h v \in V^2(T)\), and hence \[ |v - \Pi_h v|_{H^1(T)} \leq C_4(T) |v|_{H^2(T)} < K_4(T) |v|_{H^2(T)}.\tag{1} \] Now, note that \[ K_4(T) < R_T = \frac{ABC}{4S}, \] where \(R_T\) is the radius of the circumscribed circle of \(T\). Therefore, we have \[ |v - \Pi_h v|_{H^1(T)} \leq R_T |v|_{H^2(T)}.\tag{2} \] We have to realize that (1) and (2) hold for any triangles, while a geometric condition on triangles is imposed usually to obtain an error estimation in textbooks of finite element methods [S. C. Brenner and L. R. Scott, The mathematical theory of finite element methods. 3rd ed. New York, NY: Springer (2008; Zbl 1135.65042); P. G. Ciarlet, The finite element methods for elliptic problems. Philadelphia, PA: SIAM (2002; Zbl 0999.65129); A. Ern and J.-L. Guermond, Theory and practice of finite elements. New York, NY: Springer (2004; Zbl 1059.65103)].
In this paper, the author explains how these upper bounds are obtained using self-validated numerical computation. Although, the details are omitted due to page restriction, readers can see the outline of the method. It is desired that the author will publish a paper with full description of his method.
For the entire collection see [Zbl 1329.00187].

MSC:

65D05 Numerical interpolation
65N15 Error bounds for boundary value problems involving PDEs
41A05 Interpolation in approximation theory
41A63 Multidimensional problems
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65G20 Algorithms with automatic result verification

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