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High Accuracy Techniques Based Adaptive Finite Element Methods for Elliptic PDEs

Jingjing Xiao, Ying Liu, Nianyu Yi

Abstract

This paper aims to develop an efficient adaptive finite element method for the second-order elliptic problem. Although the theory for adaptive finite element methods based on residual-type a posteriori error estimator and bisection refinement has been well established, in practical computations, the use of non-asymptotic exact of error estimator and the excessive number of adaptive iteration steps often lead to inefficiency of the adaptive algorithm. We propose an efficient adaptive finite element method based on high-accuracy techniques including the superconvergence recovery technique and high-quality mesh optimization. The centroidal Voronoi Delaunay triangulation mesh optimization is embedded in the mesh adaption to provide high-quality mesh, and then assure that the superconvergence property of the recovered gradient and the asymptotical exactness of the error estimator. A tailored adaptive strategy, which could generate high-quality meshes with a target number of vertices, is developed to ensure the adaptive computation process terminated within $7$ steps. The effectiveness and robustness of the adaptive algorithm is numerically demonstrated.

High Accuracy Techniques Based Adaptive Finite Element Methods for Elliptic PDEs

Abstract

This paper aims to develop an efficient adaptive finite element method for the second-order elliptic problem. Although the theory for adaptive finite element methods based on residual-type a posteriori error estimator and bisection refinement has been well established, in practical computations, the use of non-asymptotic exact of error estimator and the excessive number of adaptive iteration steps often lead to inefficiency of the adaptive algorithm. We propose an efficient adaptive finite element method based on high-accuracy techniques including the superconvergence recovery technique and high-quality mesh optimization. The centroidal Voronoi Delaunay triangulation mesh optimization is embedded in the mesh adaption to provide high-quality mesh, and then assure that the superconvergence property of the recovered gradient and the asymptotical exactness of the error estimator. A tailored adaptive strategy, which could generate high-quality meshes with a target number of vertices, is developed to ensure the adaptive computation process terminated within steps. The effectiveness and robustness of the adaptive algorithm is numerically demonstrated.

Paper Structure

This paper contains 8 sections, 2 theorems, 33 equations, 12 figures, 5 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Model problem and adaptive finite element method.
  3. Model problem and its finite element discretization.
  4. Poor performance of the standard adaptive finite element method.
  5. High accuracy techniques based adaptive finite element method
  6. High-quality mesh generation technique
  7. Adaptive finite element algorithm
  8. Numerical results

Key Result

Theorem 2.1

There exists constant $C$ depending on the domain $\Omega$, the coefficient function $A$, and the regularity of $\mathcal{T}_h$ such that the residual type estimator is the globally upper and locally lower bounds of the error's energy norm, i.e., where $osc(f)$ is the high-order oscillation term, which is defined by $\overline{R_T(u_h)}$ denotes the element average of $R_T(u_h)$ on $T$, and $\om

Figures (12)

  • Figure 1: Example \ref{['eg1']}, (a) initial mesh; (b) numerical solution; (c) errors.
  • Figure 2: Example \ref{['eg1']}, adaptive meshes at steps: 10, 20, 30, 35, 40 and 50.
  • Figure 3: Example \ref{['eg3-1']}, adaptive meshes of steps 10, 20, 30, 40, 50, and errors.
  • Figure 4: Example \ref{['uniform-cvdt']}, mesh generated in optimization steps $1$, $2$, $50$ and $201$.
  • Figure 5: Example \ref{['uniform-cvdt']}, history of error $\|\nabla u-\nabla u_h\|$.
  • ...and 7 more figures

Theorems & Definitions (9)

  • Theorem 2.1: MKN2005
  • Example 2.1
  • Theorem 2.2: yan2001LXYC2024
  • Example 2.2
  • Example 3.1
  • Example 3.2
  • Example 4.1
  • Example 4.2: Inner layer problem
  • Example 4.3: Peak problem