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Continuous finite elements satisfying a strong discrete Miranda--Talenti identity

Dietmar Gallistl, Shudan Tian

TL;DR

The paper develops continuous but $H^2$-nonconforming finite elements in 2D and 3D that satisfy a strong discrete Miranda–Talenti inequality, enabling stable discretizations of elliptic problems in nondivergence form under Cordes and of the biharmonic equation without stabilization. The construction hinges on $C^0$-continuous spaces with $C^1$-continuity on (d-2)-faces, augmented by carefully designed bubble functions; in 2D this builds on and extends the Specht element to odd orders, while in 3D a new low-order element with degree $\ell=5$ is introduced. The resulting element families provide unisolvent degrees of freedom, provable MT property, and optimal convergence in the discrete $H^2$ seminorm, with high-order accuracy in the $L^2$ norm and favorable degrees of freedom counts in 3D. Numerical experiments in 2D and 3D validate the theoretical results and demonstrate adaptive refinement effectively resolving singularities and boundary layers.

Abstract

This article introduces continuous $H^2$-nonconforming finite elements in two and three space dimensions which satisfy a strong discrete Miranda--Talenti inequality in the sense that the global $L^2$ norm of the piecewise Hessian is bounded by the $L^2$ norm of the piecewise Laplacian. The construction is based on globally continuous finite element functions with $C^1$ continuity on the vertices (2D) or edges (3D). As an application, these finite elements are used to approximate uniformly elliptic equations in non-divergence form under the Cordes condition without additional stabilization terms. For the biharmonic equation in three dimensions, the proposed methods has less degrees of freedom than existing nonconforming schemes of the same order. Numerical results in two and three dimensions confirm the practical feasibility of the proposed schemes.

Continuous finite elements satisfying a strong discrete Miranda--Talenti identity

TL;DR

The paper develops continuous but -nonconforming finite elements in 2D and 3D that satisfy a strong discrete Miranda–Talenti inequality, enabling stable discretizations of elliptic problems in nondivergence form under Cordes and of the biharmonic equation without stabilization. The construction hinges on -continuous spaces with -continuity on (d-2)-faces, augmented by carefully designed bubble functions; in 2D this builds on and extends the Specht element to odd orders, while in 3D a new low-order element with degree is introduced. The resulting element families provide unisolvent degrees of freedom, provable MT property, and optimal convergence in the discrete seminorm, with high-order accuracy in the norm and favorable degrees of freedom counts in 3D. Numerical experiments in 2D and 3D validate the theoretical results and demonstrate adaptive refinement effectively resolving singularities and boundary layers.

Abstract

This article introduces continuous -nonconforming finite elements in two and three space dimensions which satisfy a strong discrete Miranda--Talenti inequality in the sense that the global norm of the piecewise Hessian is bounded by the norm of the piecewise Laplacian. The construction is based on globally continuous finite element functions with continuity on the vertices (2D) or edges (3D). As an application, these finite elements are used to approximate uniformly elliptic equations in non-divergence form under the Cordes condition without additional stabilization terms. For the biharmonic equation in three dimensions, the proposed methods has less degrees of freedom than existing nonconforming schemes of the same order. Numerical results in two and three dimensions confirm the practical feasibility of the proposed schemes.
Paper Structure (27 sections, 15 theorems, 83 equations, 5 figures, 6 tables)

This paper contains 27 sections, 15 theorems, 83 equations, 5 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. General notation
  4. Meshes and discrete functions
  5. An identity from integration by parts
  6. Design of the finite element in two dimensions
  7. Notation
  8. Review of the second-order Specht element
  9. Extension of the second-order Specht element to any odd order
  10. Example for $\ell=4$
  11. Example for $\ell=3$
  12. Design of the finite element on three dimensions
  13. Notation
  14. Degrees of freedom
  15. Shape functions
  16. ...and 12 more sections

Key Result

Lemma 2.1

Let $\mathcal{T}$ be a regular triangulation of $\Omega\subset \mathbb{R}^d,d=2,3$ and $V_{\mathcal{T}}\subseteq H^2(\mathcal{T})$ be a $C^0$ finite element space that has $C^1$ continuity on $(d-2)$-dimensional hyperfaces. Then any $v_h\in V_{\mathcal{T}}$ satisfies Here $(t_{F_i})_{i=1}^{d-1}$ is any orthonormal set of tangential vectors to $F\in\mathcal{F}$.

Figures (5)

  • Figure 1: Enumeration of vertices and edges in a triangle.
  • Figure 2: Mnemonic diagram of the two-dimensional finite element for $\ell=4$ (left) and $\ell=3$ (right).
  • Figure 3: Visualization of the degrees of freedom of the three-dimensional finite element for $\ell=5$.
  • Figure 4: Values of the error and $\eta$ for our element with $\ell=3$ in the second example with parameter $\iota\in \{1/100,1/10\}$.
  • Figure 5: The mesh with around $5\,127$ degrees of freedom generated by our element with $\ell=3$ when $\iota = 1/10$.

Theorems & Definitions (33)

  • Definition 1.1: strong discrete Miranda--Talenti property
  • Lemma 2.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Remark 3.3
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • ...and 23 more