A construction of canonical nonconforming finite element spaces for elliptic equations of any order in any dimension

2024/09/10 by Jia Li, Shuonan Wu, Li, Jia +1 · 1 citation
Computer Science · Engineering · #65N12 #65N30 #Advanced Mathematical Modeling in Engineering #Advanced Numerical Methods in Computational Mathematics #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods in engineering

paper · pdf · doi:10.48550/arxiv.2409.06134

openalex publication_date 2024/09/10 · openalex created_date 2024/10/22 · openalex updated_date 2026/07/28

Abstract

A unified construction of canonical H^m-nonconforming finite elements is developed for n-dimensional simplices for any m, n ≥ 1. Consistency with the Morley-Wang-Xu elements [Math. Comp. 82 (2013), pp. 25-43] is maintained when m ≤ n. In the general case, the degrees of freedom and the shape function space exhibit well-matched multi-layer structures that ensure their alignment. Building on the concept of the nonconforming bubble function, the unisolvence is established using an equivalent integral-type representation of the shape function space and by applying induction on m. The corresponding nonconforming finite element method applies to 2m-th order elliptic problems, with numerical results for m=3 and m=4 in 2D supporting the theoretical analysis.

Cited by

• Nonconforming Linear Element Method for a Generalized Tensor-Valued Stokes Equation with Application to the Triharmonic Equation

Related

• Nonconforming Finite Element Spaces for 2m-th Order Partial Differential Equations on ℝⁿ Simplicial Grids When m=n+1
• Two families of n-rectangle nonconforming finite elements for sixth-order elliptic equations
• Nonconforming finite element spaces for HΛ^k in ℝⁿ
• P₁--Nonconforming Quadrilateral Finite Element Space with Periodic Boundary Conditions: Part II. Application to the Nonconforming Heterogeneous Multiscale Method
• P_m Interior Penalty Nonconforming Finite Element Methods for 2m-th Order PDEs in ℝⁿ