Geodesic
Nonconforming P1 elements on distorted triangulations: Lower bounds for the discrete energy norm error
Applications of Mathematics, Tome 62 (2017) no. 5, pp. 433-457.

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Compared to conforming P1 finite elements, nonconforming P1 finite element discretizations are thought to be less sensitive to the appearance of distorted triangulations. E.g., optimal-order discrete $H^1$ norm best approximation error estimates for $H^2$ functions hold for arbitrary triangulations. However, the constants in similar estimates for the error of the Galerkin projection for second-order elliptic problems show a dependence on the maximum angle of all triangles in the triangulation. We demonstrate on an example of a special family of distorted triangulations that this dependence is essential, and due to the deterioration of the consistency error. We also provide examples of sequences of triangulations such that the nonconforming P1 Galerkin projections for a Poisson problem with polynomial solution do not converge or converge at arbitrarily low speed. The results complement analogous findings for conforming P1 finite elements.
DOI : 10.21136/AM.2017.0150-17
Classification : 65N12, 65N15, 65N30
Keywords: nonconforming P1 element; lowest order Raviart-Thomas element; discrete energy norm estimate; divergence of finite element method; maximum angle condition; distorted triangulation 
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Oswald, Peter. Nonconforming P1 elements on distorted triangulations: Lower bounds for the discrete energy norm error. Applications of Mathematics, Tome 62 (2017) no. 5, pp. 433-457. doi : 10.21136/AM.2017.0150-17. http://geodesic.mathdoc.fr/articles/10.21136/AM.2017.0150-17/

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