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Lowest-degree robust finite element schemes for inhomogeneous bi-Laplace problems

Bin Dai, Huilan Zeng, Chensong Zhang, Shuo Zhang

TL;DR

The paper introduces a lowest-degree robust finite element approach on rectangular grids for inhomogeneous bi-Laplace problems, treating both a fourth-order elliptic perturbation and the Helmholtz transmission eigenvalue problem. It leverages the reduced rectangular Morley space $V_h^{\mathrm{R}}$ with piecewise quadratic polynomials and proves a discrete Grisvard-type identity, complemented by a locally-averaged interpolation that reproduces $P_{2}$ locally and yields optimal error bounds independent of $\varepsilon$. A novel extended-grid interpolator and an $L^1$-based, non-projection operator $\Pi_{h0}$ enable sharp broken $H^1$ and $H^2$ approximations on convex and non-convex domains. The resulting robust schemes achieve linear convergence in the energy-like norm for the perturbation problem and quadratic convergence for Helmholtz eigenvalues, with numerical experiments confirming the theoretical rates and demonstrating stability under coefficient inhomogeneity and grid variation.

Abstract

In this paper, we study the numerical method for the bi-Laplace problems with inhomogeneous coefficients; particularly, we propose finite element schemes on rectangular grids respectively for an inhomogeneous fourth-order elliptic singular perturbation problem and for the Helmholtz transmission eigenvalue problem. The new methods use the reduced rectangle Morley (RRM for short) element space with piecewise quadratic polynomials, which are of the lowest degree possible. For the finite element space, a discrete analogue of an equality by Grisvard is proved for the stability issue and a locally-averaged interpolation operator is constructed for the approximation issue. Optimal convergence rates of the schemes are proved, and numerical experiments are given to verify the theoretical analysis.

Lowest-degree robust finite element schemes for inhomogeneous bi-Laplace problems

TL;DR

The paper introduces a lowest-degree robust finite element approach on rectangular grids for inhomogeneous bi-Laplace problems, treating both a fourth-order elliptic perturbation and the Helmholtz transmission eigenvalue problem. It leverages the reduced rectangular Morley space with piecewise quadratic polynomials and proves a discrete Grisvard-type identity, complemented by a locally-averaged interpolation that reproduces locally and yields optimal error bounds independent of . A novel extended-grid interpolator and an -based, non-projection operator enable sharp broken and approximations on convex and non-convex domains. The resulting robust schemes achieve linear convergence in the energy-like norm for the perturbation problem and quadratic convergence for Helmholtz eigenvalues, with numerical experiments confirming the theoretical rates and demonstrating stability under coefficient inhomogeneity and grid variation.

Abstract

In this paper, we study the numerical method for the bi-Laplace problems with inhomogeneous coefficients; particularly, we propose finite element schemes on rectangular grids respectively for an inhomogeneous fourth-order elliptic singular perturbation problem and for the Helmholtz transmission eigenvalue problem. The new methods use the reduced rectangle Morley (RRM for short) element space with piecewise quadratic polynomials, which are of the lowest degree possible. For the finite element space, a discrete analogue of an equality by Grisvard is proved for the stability issue and a locally-averaged interpolation operator is constructed for the approximation issue. Optimal convergence rates of the schemes are proved, and numerical experiments are given to verify the theoretical analysis.
Paper Structure (28 sections, 17 theorems, 104 equations, 15 figures, 2 tables)

This paper contains 28 sections, 17 theorems, 104 equations, 15 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Inhomogeneous fourth order elliptic perturbation problem
  4. Helmholtz transmission eigenvalue problem
  5. Subdivisions and finite elements
  6. An optimal interpolator to the reduced rectangular Morley element space
  7. Reduced rectangular Morley element space revisited
  8. An auxiliary interpolator on an extended grid
  9. Extension of the grid
  10. An auxiliary interpolator
  11. An optimal interpolator to reduced rectangular Morley element space
  12. Proof of Theorem \ref{['thm:approxH02']}
  13. Reduced rectangular Morley element schemes for model problems
  14. Robust RRM scheme for fourth order elliptic perturbation problem
  15. Optimal scheme for Helmholtz transmission eigenvalue problem
  16. ...and 13 more sections

Key Result

Lemma 2.1

( XY.Meng;XQ.Yang;S.Zhang2016) For any function $v_{h} \in V_{hs}^{\rm{M}}$, we have the following estimates:

Figures (15)

  • Figure 1: Illustration of a $3 \times 3$ patch $\mathcal{M}_{K}$.
  • Figure 2: Illustration of a $5 \times 5$ patch centered at $K$.
  • Figure 3: Left: Illustration of $5 \times 5$ patch $F_{K_1}$. Right: Illustration of the basis function $\varphi_{K_1}$ and $\varphi_{K_2}$
  • Figure 4: Expansion outside $\Omega$ and four basis functions added corresponding to a convex corner node (left) and a concave corner node (right).
  • Figure 5: Expansion outside a non-corner boundary edge (left or right) and two basis functions added.
  • ...and 10 more figures

Theorems & Definitions (34)

  • Lemma 2.1
  • Lemma 3.1
  • Definition 3.2
  • Remark 3.3
  • Proposition 3.4
  • Lemma 3.5
  • Proposition 3.6
  • Remark 3.7
  • Lemma 3.8: discrete analogue of \ref{['eq:Grisvard']}
  • proof
  • ...and 24 more