A Note on the Shape Regularity of Worsey-Farin Splits

Article Mathematics, Applied

Convergence of Lagrange finite elements for the Maxwell eigenvalue problem in two dimensions

Daniele Boffi et al.

Summary: This article examines finite element approximations of the two-dimensional Maxwell eigenvalue problem, proves the convergence of discrete eigenvalues using Lagrange finite elements in certain cases, and provides numerical experiments to support the theoretical results. The computations also reveal that the eigenvalue approximations are highly sensitive to nearly singular vertices on general triangulations.

IMA JOURNAL OF NUMERICAL ANALYSIS (2023)

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Article Engineering, Multidisciplinary

Low-order divergence-free approximations for the Stokes problem on Worsey-Farin and Powell-Sabin splits

Maurice Fabien et al.

Summary: This paper presents low-order, inf-sup stable, and divergence-free finite element approximations for the Stokes problem, using Worsey-Farin splits in three dimensions and Powell-Sabin splits in two dimensions. The velocity space consists of continuous, piecewise linear polynomials, while the pressure space is a subspace of piecewise constants with weak continuity properties at singular edges (3D) and singular vertices (2D). Implementation aspects related to coding the pressure space are discussed, showing that pressure constraints can be enforced at an algebraic level.

COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING (2022)

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Article Mathematics, Applied

EXACT SEQUENCES ON WORSEY-FARIN SPLITS

Johnny Guzman et al.

Summary: This paper introduces smooth finite element spaces defined on three-dimensional Worsey-Farin splits, constructing C-1, H-1(curl), and H-1-conforming finite element spaces that satisfy local exactness properties.

MATHEMATICS OF COMPUTATION (2022)

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Article Mathematics, Applied

Partially Discontinuous Nodal Finite Elements for H(curl) and H(div)

Jun Hu et al.

Summary: This paper investigates the discretization of H(curl) and H(div) in two and three space dimensions using partially discontinuous nodal finite elements, which are vector-valued Lagrange finite elements with discontinuity in certain directions. These spaces can be implemented as a combination of continuous and discontinuous Lagrange elements and fit in de Rham complexes. Well-conditioned nodal bases are constructed.

COMPUTATIONAL METHODS IN APPLIED MATHEMATICS (2022)

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Article Mathematics, Applied