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On the best accuracy using the $h$-adaptive finite element refinement

Jie Liu

Abstract

In \cite{liu2022practical}, a general algorithm is developed to efficiently obtain the best accuracy using the regular refinement. The adaptive refinement allows for obtaining an accuracy with a smaller number of DoFs compared with the regular refinement. In this paper, we investigate the best accuracy when using the adaptive refinement. To this end, we study the evolution of the truncation error and the round-off error using the adaptive refinement. For the former, a new threshold for the selection of the number of elements to be refined is proposed. For the latter, the round-off error is quantified using the method proposed in \cite{liu2022practical}. Moreover, for achieving a tolerance, we propose to use the line of the round-off error as a stopping criterion.

On the best accuracy using the $h$-adaptive finite element refinement

Abstract

In \cite{liu2022practical}, a general algorithm is developed to efficiently obtain the best accuracy using the regular refinement. The adaptive refinement allows for obtaining an accuracy with a smaller number of DoFs compared with the regular refinement. In this paper, we investigate the best accuracy when using the adaptive refinement. To this end, we study the evolution of the truncation error and the round-off error using the adaptive refinement. For the former, a new threshold for the selection of the number of elements to be refined is proposed. For the latter, the round-off error is quantified using the method proposed in \cite{liu2022practical}. Moreover, for achieving a tolerance, we propose to use the line of the round-off error as a stopping criterion.

Paper Structure

This paper contains 25 sections, 12 equations, 8 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Model problem and FEM formulation
  3. Model problem
  4. FEM formulation
  5. Grid generation
  6. Error estimation
  7. The optimal mesh for achieving a tolerance
  8. Error reduction using the AMR strategy
  9. A new method for seeking pct
  10. Step 1: selecting pctinit
  11. Step 2: seeking pctopt
  12. Results
  13. Round-off error in h-adaptive refinement
  14. Using different pct on the same mesh
  15. Using one constant pct to refine one mesh continually
  16. ...and 10 more sections

Figures (8)

  • Figure 1: Shapes of the 2D geometry.
  • Figure 2: Shapes of the 2D solution on the unit square.
  • Figure 3: Influence of the pct on the error.
  • Figure 4: Illustration of the procedure for seeking pct$_{\rm opt}$.
  • Figure 5: Initial mesh for the problem of Eq. (\ref{['equation_2d_general']}) with $c=1e$-5.
  • ...and 3 more figures