Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Smoother-type a posteriori error estimates for finite element methods

Yuwen Li, Han Shui

TL;DR

This work develops user-friendly a posteriori error estimates of finite element methods, based on smoothers of linear iterative solvers, which outperform residual-type estimators in accuracy and exhibit robustness with respect to parameters and polynomial degrees.

Abstract

This work develops user-friendly a posteriori error estimates of finite element methods, based on smoothers of linear iterative solvers. The proposed method employs simple smoothers, such as Jacobi or Gauss-Seidel iteration, on an auxiliary finer mesh to process the finite element residual for a posteriori error control. The implementation has linear complexity and requires only a coarse-to-fine prolongation operator. For symmetric problems, we prove the reliability and efficiency of smoother-type error estimators under a saturation assumption. Numerical experiments for various PDEs demonstrate that the proposed smoother-type error estimators outperform residual-type estimators in accuracy and exhibit robustness with respect to parameters and polynomial degrees.

Smoother-type a posteriori error estimates for finite element methods

TL;DR

This work develops user-friendly a posteriori error estimates of finite element methods, based on smoothers of linear iterative solvers, which outperform residual-type estimators in accuracy and exhibit robustness with respect to parameters and polynomial degrees.

Abstract

This work develops user-friendly a posteriori error estimates of finite element methods, based on smoothers of linear iterative solvers. The proposed method employs simple smoothers, such as Jacobi or Gauss-Seidel iteration, on an auxiliary finer mesh to process the finite element residual for a posteriori error control. The implementation has linear complexity and requires only a coarse-to-fine prolongation operator. For symmetric problems, we prove the reliability and efficiency of smoother-type error estimators under a saturation assumption. Numerical experiments for various PDEs demonstrate that the proposed smoother-type error estimators outperform residual-type estimators in accuracy and exhibit robustness with respect to parameters and polynomial degrees.

Paper Structure

This paper contains 17 sections, 4 theorems, 79 equations, 16 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Abstract Framework
  3. Abstract Error Estimators
  4. Application to the Poisson Equation
  5. Discrete Ambient Space
  6. Practical Estimators in H(grad)
  7. Pointwise Smoother-type Error Estimators
  8. Localized Error Estimator and Adaptive Algorithm
  9. Numerical Experiments in H(grad)
  10. A Posteriori Error Estimates in H(curl)
  11. Block Diagonal Estimators for Symmetric Systems
  12. Poisson in Mixed Form
  13. Biot's Consolidation Model
  14. Error Estimators for Non-symmetric Problems
  15. Convection-diffusion equation
  16. ...and 2 more sections

Key Result

Theorem 2.1

Let $B_a=I_hA_h^{-1}I_h^*+S_a: V^*\rightarrow V$ with $S_a:=\sum_{k=1}^NI_kA_k^{-1}I_k^*$. Assume that $B_a$ is a preconditioner for the SPD operator $A$, i.e., where $c_0, c_1>0$ are uniform constants. Then with $r=f-Au_h$, it holds that

Figures (16)

  • Figure 3.1: Assignment of $\langle r_{h/2},S^a_{h/2}r_{h/2}\rangle$ to each element in the refinement $\mathcal{T}_{h/2}$.
  • Figure 3.2: Convergence of error estimators based on fine-coarse (left) and $\mathcal{P}_2$-$\mathcal{P}_1$ (right) layers.
  • Figure 3.3: Convergence of $\mathcal{P}_p$-AFEMs driven by pointwise smoother-type error estimators.
  • Figure 3.4: Adaptive error estimator convergence in the $L^2$ norm for the Poisson equation in fine-coarse layers (left) and $\mathcal{P}_2$-$\mathcal{P}_1$ layers (right), where $N$ is the number of triangles.
  • Figure 3.5: Convergence of error estimators for $H({\rm curl})$ problems in 2D (left) and 3D (right).
  • ...and 11 more figures

Theorems & Definitions (12)

  • Theorem 2.1
  • proof
  • Theorem 2.2
  • proof
  • Remark 2.3
  • Corollary 2.4
  • proof
  • Remark 3.1
  • Remark 3.2: Inhomogeneous essential boundary condition in H(grad)
  • Remark 3.3
  • ...and 2 more