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The Zipped Finite Element Method: High-order Shape Functions for Polygons

Stefano Berrone, Lorenzo Neva, Moreno Pintore, Gioana Teora, Fabio Vicini

Abstract

In this paper, we present a new polygonal finite element method, called the Zipped Finite Element Method, for star-shaped polygons. The proposed approach constructs high-order shape functions as linear combinations of standard finite element basis functions defined on a local trivial sub-triangulation of each element. This refinement is used solely for the construction of the shape functions and does not affect the final number of degrees of freedom. The resulting finite element space includes polynomials of the desired order and preserves conformity across elements. Consequently, the method inherits the convergence properties of the finite element framework under suitable mesh assumptions. Numerical experiments confirm the expected rates of convergence.

The Zipped Finite Element Method: High-order Shape Functions for Polygons

Abstract

In this paper, we present a new polygonal finite element method, called the Zipped Finite Element Method, for star-shaped polygons. The proposed approach constructs high-order shape functions as linear combinations of standard finite element basis functions defined on a local trivial sub-triangulation of each element. This refinement is used solely for the construction of the shape functions and does not affect the final number of degrees of freedom. The resulting finite element space includes polynomials of the desired order and preserves conformity across elements. Consequently, the method inherits the convergence properties of the finite element framework under suitable mesh assumptions. Numerical experiments confirm the expected rates of convergence.

Paper Structure

This paper contains 12 sections, 4 theorems, 58 equations, 5 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Notations and the Model problem
  3. The Zipped Finite Element Space
  4. High-order shape functions on polygons
  5. Standard basis functions over the element sub-triangulation
  6. Selection of the nodal coordinates
  7. The determination of shape functions
  8. The Zipped Discrete Problem
  9. Numerical Experiments
  10. Test 1: Polynomials Approximation
  11. Test 2: Convergence rates
  12. Conclusion

Key Result

Theorem 1

Suppose $\{L_0,\dots,L_k\}$ are $k+1$ distinct lines on $\mathbb{R}^2$ and $U = \{\bm{x}_1, \dots, \bm{x}_{n_k}\}$ is a set of $n_k$ distinct points such that $\bm{x}_1 \in L_0$, $\bm{x}_2, \bm{x}_3 \in L_1\setminus L_0$, $\dots$, and $\bm{x}_{n_k - k},\dots,\bm{x}_{n_k} \in L_k \setminus \{L_0,\dot

Figures (5)

  • Figure 1: Different sub-triangulations of a heptagon. Left: Sub-triangulation obtained by defining $\bm{x}^E$ (the red dot) as the solution of \ref{['eq:opt:centerstarshaped']}. Right: Sub-triangulation obtained by defining $\bm{x}^E$ (the green dot) as the solution of \ref{['eq:opt:ssa']}. The black line denotes the circumference of the ball with respect to which the heptagon is star-shaped.
  • Figure 2: Selection of local $N_{\operatorname{dof}}\ifstrempty{E}{}{^{E}}$ DOFs on a triangular element $E$ for different orders $k$. Red dots indicate $\mathcal{C}\ifstrempty{E}{}{^{E}}$, while blue squares represent $\mathcal{K}\ifstrempty{E}{}{^{E}}$. The yellow lines on $k=4$ represent the distinct lines $\{L_0, \dots, L_k\}$ defined in Theorem \ref{['theor:ChungYao']}.
  • Figure 3: Selection of local $N_{\operatorname{dof}}\ifstrempty{E}{}{^{E}}$ DOFs with the proposed heuristic procedure on a convex octagon $E$. Red dots indicate $\mathcal{C}\ifstrempty{E}{}{^{E}}$, while blue squares represent $\mathcal{K}\ifstrempty{E}{}{^{E}}$. Left: $k=4$. Right: $k=6$.
  • Figure 4: Test 2: Last refinement for each family of meshes employed.
  • Figure 5: Test 2: Behavior of errors \ref{['eq:errors']} as $N_{\operatorname{dof}}\ifstrempty{}{}{^{}}$ increases. Each column corresponds to a different family of meshes. Left: Random Distorted. Center: Voronoi. Right: Structured Concave. Each row refers to a different value of $k = 1, 3, 5$. Solid lines: $L^{2}\ifstrempty{}{}{_{}}\ifstrempty{}{}{\!\left(\right)}$-errors. Dashed lines: $H^{1}\ifstrempty{}{}{_{}}\ifstrempty{}{}{\!\left(\right)}$-errors.

Theorems & Definitions (9)

  • Remark 1
  • Theorem 1: Theorem 6.1 in Gregory2007
  • Proposition 1
  • proof
  • Remark 2
  • Proposition 2
  • proof
  • Proposition 3
  • proof