Nonconforming P1 elements on distorted triangulations: lower bounds for the discrete energy norm error.

DOI10.21136/AM.2017.0150-17zbMATH Open1488.65650arXiv1607.05983MaRDI QIDQ1686813

Publication date: 15 December 2017

Published in: Applications of Mathematics (Search for Journal in Brave)

Abstract: 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, 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 the 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 slow speed. The results complement analogous findings for conforming P1 elements.

Full work available at URL: https://arxiv.org/abs/1607.05983

zbMATH Keywords

maximum angle condition discrete energy norm estimate distorted triangulation divergence of finite element method lowest order Raviart-Thomas element nonconforming P1 element

Mathematics Subject Classification ID

Error bounds for boundary value problems involving PDEs (65N15) Stability and convergence of numerical methods for boundary value problems involving PDEs (65N12) Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs (65N30) Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs (65N50)