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Two families of C1-Pk Fraeijs de Veubeke-Sander finite elements on quadrilateral meshes

Shangyou Zhang

TL;DR

This work extends the $C^1$-$P_k$ Fraeijs de Veubeke-Sander finite elements to general quadrilateral meshes by subdividing each quadrilateral into four triangles and using four $P_k$ pieces per quad. It introduces two global finite element spaces: a full space $V_h^{(1)}$ on the macro-mesh and a condensed space $V_h^{(2)}$ with edge-dof condensation, both enabling $C^1$ continuity across quadrilaterals. Uni-solvency is established via a key lemma on triangular patches, and optimal convergence is proven for $k\ge 3$ with the error bound $|u-u_h|_0 + h|u-u_h|_1 + h^2|u-u_h|_2 \le C h^{k+1} |u|_{k+1}$. Numerical tests show that the full-space elements retain optimal convergence for interface problems, while condensed elements offer higher accuracy in smooth cases but may degrade to $O(h^{1/2})$ when coefficient jumps occur, highlighting a trade-off between degrees of freedom and robustness.

Abstract

We extend the $C^1$-$P_3$ Fraeijs de Veubeke-Sander finite element to two families of $C^1$-$P_k$ ($k>3$) macro finite elements on general quadrilateral meshes. On each quadrilateral, four $P_k$ polynomials are defined on the four triangles subdivided from the quadrilateral by its two diagonal lines. The first family of $C^1$-$P_k$ finite elements is the full $C^1$-$P_k$ space on the macro-mesh. Thus the element can be applied to interface problems. The second family of $C^1$-$P_k$ finite elements condenses all internal degrees of freedom by moving them to the four edges. Thus the second element method has much less unknowns but is more accurate than the first one. We prove the uni-solvency and the optimal order convergence. Numerical tests and comparisons with the $C^1$-$P_k$ Argyris are provided.

Two families of C1-Pk Fraeijs de Veubeke-Sander finite elements on quadrilateral meshes

TL;DR

This work extends the - Fraeijs de Veubeke-Sander finite elements to general quadrilateral meshes by subdividing each quadrilateral into four triangles and using four pieces per quad. It introduces two global finite element spaces: a full space on the macro-mesh and a condensed space with edge-dof condensation, both enabling continuity across quadrilaterals. Uni-solvency is established via a key lemma on triangular patches, and optimal convergence is proven for with the error bound . Numerical tests show that the full-space elements retain optimal convergence for interface problems, while condensed elements offer higher accuracy in smooth cases but may degrade to when coefficient jumps occur, highlighting a trade-off between degrees of freedom and robustness.

Abstract

We extend the - Fraeijs de Veubeke-Sander finite element to two families of - () macro finite elements on general quadrilateral meshes. On each quadrilateral, four polynomials are defined on the four triangles subdivided from the quadrilateral by its two diagonal lines. The first family of - finite elements is the full - space on the macro-mesh. Thus the element can be applied to interface problems. The second family of - finite elements condenses all internal degrees of freedom by moving them to the four edges. Thus the second element method has much less unknowns but is more accurate than the first one. We prove the uni-solvency and the optimal order convergence. Numerical tests and comparisons with the - Argyris are provided.

Paper Structure

This paper contains 4 sections, 3 theorems, 6 equations, 5 figures, 8 tables.

Table of Contents

  1. Introduction
  2. The $C^1$-$P_k$ Fraeijs deVeubeke and Sanderfinite elements
  3. Uni-solvency and convergence
  4. Numerical Experiments

Key Result

Lemma 3.1

Let a triangle $\b x_1\b x_2\b x_4$ be subdivided into two triangles $T_1=\b x_1\b x_2\b x_0$ and $T_4=\b x_1\b x_0\b x_4$, cf. Figure 1-e0. Let the $C^1$-$P_k$ function $p$ satisfy, for any $0<m\le k-2$, where $\partial_{(\b x_1\b x_2^\perp)^i} p_1$ is the $i$-th normal derivative to the edge. Then, the $m+1$ tangential derivative at $\b x_1$ vanishes, i.e., ∂_(x̱_1x̱_0)^m+1 p_1(x̱_0) = 0.

Figures (5)

  • Figure 1: The 16, 28 and 44 degrees of freedom of the $C^1$-$P_3$, $P_4$ and $P_5$ Fraeijs deVeubeke and Sander macro-elements.
  • Figure 2: A triangle $\b x_1\b x_2\b x_4$ at the origin $\b x_1(0,0)$ is split into two by the horizontal line $\b x_1\b x_0$.
  • Figure 3: A quadrilateral $Q=\b x_1\b x_2\b x_3\b x_4$ is subdivided into four triangles by its two diagonals.
  • Figure 4: The first three grids for the computation in Tables \ref{['t-1']}, \ref{['t-3']}, \ref{['t-4']}, \ref{['t-6']} and \ref{['t-7']}.
  • Figure 5: The first three levels of grids for the computation in Tables \ref{['t-2']}, \ref{['t-5']} and \ref{['t-8']}.

Theorems & Definitions (6)

  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Theorem 3.3
  • proof