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A high order stabilization-free virtual element method for general second-order elliptic eigenvalue problem

Liangkun Xu, Shixi Wang, Yidu Yang, Hai Bi

Abstract

In this paper, we discuss a novel higher-order stabilization-free virtual element method for general second-order elliptic eigenvalue problems. Optimal a priori error estimates are derived for both the approximate eigenspace and eigenvalues. Numerical experiments are conducted on regular convex polygonal meshes, convex-concave polygonal meshes, and concave polygonal meshes. The numerical results validate the effectiveness of the proposed method.

A high order stabilization-free virtual element method for general second-order elliptic eigenvalue problem

Abstract

In this paper, we discuss a novel higher-order stabilization-free virtual element method for general second-order elliptic eigenvalue problems. Optimal a priori error estimates are derived for both the approximate eigenspace and eigenvalues. Numerical experiments are conducted on regular convex polygonal meshes, convex-concave polygonal meshes, and concave polygonal meshes. The numerical results validate the effectiveness of the proposed method.

Paper Structure

This paper contains 4 sections, 3 theorems, 48 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. Eigenvalue problems and their high-order stabilization-free virtual element approximations
  3. A priori error analysis
  4. Numerical experiments

Key Result

Lemma 1

Let $f \in H^{\max\{0,s-1\}}(\Omega)$, and $w \in H^{1+s}(\Omega) \cap H_0^1(\Omega)$$(0 \leq r\leq s \leq k)$ be the solution of a2.4. Then the unique solution $w_h$ of a3.14 satisfies the error estimate furthermore, it holds that $\blacktriangleleft$$\blacktriangleleft$

Figures (8)

  • Figure 1: Different polygonal discretizations of the unit square.
  • Figure 2: In Case 1, the error curves for $k=2$: left ($\mathcal{T}_h^1$), middle ($\mathcal{T}_h^2$), right ($\mathcal{T}_h^3$).
  • Figure 3: In Case 1, the error curves for $k=3$: left ($\mathcal{T}_h^1$), middle ($\mathcal{T}_h^2$), right ($\mathcal{T}_h^3$).
  • Figure 4: In Case 1, the error curves for $k=4$: left ($\mathcal{T}_h^1$), middle ($\mathcal{T}_h^2$), right ($\mathcal{T}_h^3$).
  • Figure 5: In Case 2, the error curves for $k=2$: left ($\mathcal{T}_h^1$), middle ($\mathcal{T}_h^2$), right ($\mathcal{T}_h^3$).
  • ...and 3 more figures

Theorems & Definitions (5)

  • Lemma 1
  • Remark 1
  • Proposition 1
  • Theorem 3.1
  • proof