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Superclosenes error estimates for the div least-squares finite element method on elliptic problems

Gang Chen, Fanyi Yang, Zheyuan Zhang

Abstract

In this paper we provide some error estimates for the div least-squares finite element method on elliptic problems. The main contribution is presenting a complete error analysis, which improves the current \emph{state-of-the-art} results. The error estimates for both the scalar and the flux variables are established by specially designed dual arguments with the help of two projections: elliptic projection and H(div) projection, which are crucial to supercloseness estimates. In most cases, $H^3$ regularity is omitted to get the optimal convergence rate for vector and scalar unknowns, and most of our results require a lower regularity for the vector variable than the scalar. Moreover, a series of supercloseness results are proved, which are \emph{never seen} in the previous work of least-squares finite element methods.

Superclosenes error estimates for the div least-squares finite element method on elliptic problems

Abstract

In this paper we provide some error estimates for the div least-squares finite element method on elliptic problems. The main contribution is presenting a complete error analysis, which improves the current \emph{state-of-the-art} results. The error estimates for both the scalar and the flux variables are established by specially designed dual arguments with the help of two projections: elliptic projection and H(div) projection, which are crucial to supercloseness estimates. In most cases, regularity is omitted to get the optimal convergence rate for vector and scalar unknowns, and most of our results require a lower regularity for the vector variable than the scalar. Moreover, a series of supercloseness results are proved, which are \emph{never seen} in the previous work of least-squares finite element methods.
Paper Structure (18 sections, 21 theorems, 143 equations, 6 figures, 6 tables)

This paper contains 18 sections, 21 theorems, 143 equations, 6 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Notation
  3. Least-Squares finite element methods
  4. Primary error estimates
  5. The spaces
  6. Projection
  7. $L^2$ projection
  8. Elliptic projection
  9. Error estimates
  10. Error estimates from dual arguments
  11. Error estimates for $\Pi_{V_h}u-u_h$
  12. Error estimates for $\nabla\cdot(\mathbf\Pi_{\mathbf P_h}\mathbf q-\mathbf q_h)$
  13. Error estimates for $\mathbf\Pi_{\mathbf P_h}\mathbf q-\mathbf q_h$
  14. Error estimates for $\nabla(\Pi_{V_h}u-u_h)$
  15. Convergence tests
  16. ...and 3 more sections

Key Result

lemma thmcounterlemma

The continuity property and the coercivity holds

Figures (6)

  • Figure 1: Results for $\bm{\mathcal{RT}}$ elements
  • Figure 2: Results for $\bm{\mathcal{BDM}}$ elements
  • Figure 3: Results for $\bm{\mathcal{BDM}}_2/\mathcal{P}_2$ elements
  • Figure 4: Results for $\bm{\mathcal{RT}}$ elements
  • Figure 5: Results for $\bm{\mathcal{BDM}}$ elements
  • ...and 1 more figures

Theorems & Definitions (51)

  • remark thmcounterremark
  • lemma thmcounterlemma: MR1302685,Zhang2023
  • lemma thmcounterlemma: Orthogonality
  • lemma thmcounterlemma: Stability
  • lemma thmcounterlemma: Boffi2008MixedFE
  • lemma thmcounterlemma
  • remark thmcounterremark
  • remark thmcounterremark
  • lemma thmcounterlemma
  • lemma thmcounterlemma
  • ...and 41 more