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Towards $C^0$ finite element methods for fourth-order elliptic equation. Part I: general boundary conditions

Xihao Zhang, Hengguang Li, Nianyu Yi, Peimeng Yin

TL;DR

The paper develops a $C^0$ finite element framework for the biharmonic equation on polygonal domains with general boundary conditions by introducing a modified mixed formulation that decouples the problem into Poisson systems while enforcing the correct Sobolev space via an orthogonal decomposition of the Laplacian image. A construction of $L^2$ basis functions $ξ_m$ for the orthogonal complement and a Gram system to determine correction coefficients yields an equivalent, well-posed reformulation; a practical $C^0$-FEM is then built around a four-step algorithm that uses standard Poisson solves and a small linear system. Theoretical results provide well-posedness and error estimates for the auxiliary and primary unknowns, with convergence rates that depend on the largest interior angle and boundary types; extensive numerical tests on various polygonal domains validate the method and show it outperforms naive decouplings, including Neumann and mixed BC cases. Overall, the method enables robust, efficient, and provably convergent $C^0$ discretizations for fourth-order problems without requiring $C^1$ elements, broadening applicability to nonconvex polygons and complex boundary conditions.

Abstract

This paper is part of a series developing $C^0$ finite element methods for fourth-order elliptic equations on polygonal domains. Here, we investigate how boundary conditions influence the design of effective $C^0$ schemes, specifically focusing on equations without lower-order terms, namely the biharmonic equation. We propose a modified mixed formulation that decomposes the problem into a system of Poisson equations, where the number of equations depends on both the largest interior angle and the boundary conditions on its two adjacent sides. In contrast to the naive mixed formulation, which involves only two Poisson problems, the proposed approach guarantees convergence to the true solution for arbitrary polygonal domains and general boundary conditions, including Navier, Neumann, and mixed boundary conditions. $C^0$ finite element algorithms are developed, rigorous error estimates are established, and numerical experiments are presented to demonstrate the well-posedness and effectiveness of the proposed method.

Towards $C^0$ finite element methods for fourth-order elliptic equation. Part I: general boundary conditions

TL;DR

The paper develops a finite element framework for the biharmonic equation on polygonal domains with general boundary conditions by introducing a modified mixed formulation that decouples the problem into Poisson systems while enforcing the correct Sobolev space via an orthogonal decomposition of the Laplacian image. A construction of basis functions for the orthogonal complement and a Gram system to determine correction coefficients yields an equivalent, well-posed reformulation; a practical -FEM is then built around a four-step algorithm that uses standard Poisson solves and a small linear system. Theoretical results provide well-posedness and error estimates for the auxiliary and primary unknowns, with convergence rates that depend on the largest interior angle and boundary types; extensive numerical tests on various polygonal domains validate the method and show it outperforms naive decouplings, including Neumann and mixed BC cases. Overall, the method enables robust, efficient, and provably convergent discretizations for fourth-order problems without requiring elements, broadening applicability to nonconvex polygons and complex boundary conditions.

Abstract

This paper is part of a series developing finite element methods for fourth-order elliptic equations on polygonal domains. Here, we investigate how boundary conditions influence the design of effective schemes, specifically focusing on equations without lower-order terms, namely the biharmonic equation. We propose a modified mixed formulation that decomposes the problem into a system of Poisson equations, where the number of equations depends on both the largest interior angle and the boundary conditions on its two adjacent sides. In contrast to the naive mixed formulation, which involves only two Poisson problems, the proposed approach guarantees convergence to the true solution for arbitrary polygonal domains and general boundary conditions, including Navier, Neumann, and mixed boundary conditions. finite element algorithms are developed, rigorous error estimates are established, and numerical experiments are presented to demonstrate the well-posedness and effectiveness of the proposed method.
Paper Structure (13 sections, 14 theorems, 89 equations, 13 figures, 10 tables, 1 algorithm)

This paper contains 13 sections, 14 theorems, 89 equations, 13 figures, 10 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Biharmonic problem with mixed boundary conditions
  3. Naive mixed formulation
  4. Modified mixed formulation
  5. L^2 basis function
  6. Orthogonal space of the Laplace operator image
  7. Modified mixed formulation and its well-posedness
  8. C^0 finite element method
  9. Neumann boundary conditions
  10. Modified mixed formulation
  11. Numerical Tests
  12. Mixed boundary conditions with
  13. Neumann boundary conditions

Key Result

Lemma 2.4

grisvard1992singularities Let $\Omega$ be an open polygonal subset of $\mathbb R^2$. Then the mapping which is defined for $H^2(\Omega)$ has an unique continuous extension as an operator from $D(\Delta, L^2(\Omega))$ into $\widetilde{H}^{-{1}/{2}}(\Gamma_j) \times \widetilde{H}^{-{3}/{2}}(\Gamma_j)$.

Figures (13)

  • Figure 1: $\Omega$ and $K_\omega^R$.
  • Figure 1: Four different polygonal domains used in the numerical experiments. The re-entrant corner at point $Q$ has interior angle $\omega$. I: $\omega=\pi$, II: $\omega=5\pi/4$, III: $\omega=3\pi/2$, IV: $\omega=7\pi/4$.
  • Figure 2: \ref{['exa.6.1.180']}: Initial mesh and mesh after $1$ refinement.
  • Figure 3: \ref{['exa.6.1.180']}: Domain I; Boundary type: $B_3, B_4$; $u_h^N, u_h^M$ and their differences with $u_R$.
  • Figure 4: \ref{['ex6.1.225']}: Initial mesh and mesh after $1$ refinement.
  • ...and 8 more figures

Theorems & Definitions (33)

  • Definition 2.1: Vertex, edge, and indicator sets
  • Definition 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Proof 1
  • Lemma 2.6
  • Proof 2
  • Lemma 2.7
  • Corollary 2.8
  • Proof 3
  • ...and 23 more