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Stopping Criteria for the Conjugate Gradient Algorithm in High-Order Finite Element Methods

Yichen Guo, Eric de Sturler, Tim Warburton

TL;DR

Two new stopping criteria that balance algebraic and discretization errors for the conjugate gradient algorithm applied to high-order finite element discretizations of Poisson problems and a new error indicator derived from a recovery-based error estimator that is less computationally expensive and more reliable are introduced.

Abstract

We consider stopping criteria that balance algebraic and discretization errors for the conjugate gradient algorithm applied to high-order finite element discretizations of Poisson problems. Firstly, we introduce a new stopping criterion that suggests stopping when the norm of the linear system residual is less than a small fraction of an error indicator derived directly from the residual. This indicator shares the same mesh size and polynomial degree scaling as the norm of the residual, resulting in a robust criterion regardless of the mesh size, the polynomial degree, and the shape regularity of the mesh. Secondly, for solving Poisson problems with highly variable piecewise constant coefficients, we introduce a subdomain-based criterion that recommends stopping when the norm of the linear system residual restricted to each subdomain is smaller than the corresponding indicator also restricted to that subdomain. Reliability and efficiency theorems for the first criterion are established. Numerical experiments, including tests with highly variable piecewise constant coefficients and a GPU-accelerated three-dimensional elliptic solver, demonstrate that the proposed criteria efficiently avoid both premature termination and over-solving.

Stopping Criteria for the Conjugate Gradient Algorithm in High-Order Finite Element Methods

TL;DR

Two new stopping criteria that balance algebraic and discretization errors for the conjugate gradient algorithm applied to high-order finite element discretizations of Poisson problems and a new error indicator derived from a recovery-based error estimator that is less computationally expensive and more reliable are introduced.

Abstract

We consider stopping criteria that balance algebraic and discretization errors for the conjugate gradient algorithm applied to high-order finite element discretizations of Poisson problems. Firstly, we introduce a new stopping criterion that suggests stopping when the norm of the linear system residual is less than a small fraction of an error indicator derived directly from the residual. This indicator shares the same mesh size and polynomial degree scaling as the norm of the residual, resulting in a robust criterion regardless of the mesh size, the polynomial degree, and the shape regularity of the mesh. Secondly, for solving Poisson problems with highly variable piecewise constant coefficients, we introduce a subdomain-based criterion that recommends stopping when the norm of the linear system residual restricted to each subdomain is smaller than the corresponding indicator also restricted to that subdomain. Reliability and efficiency theorems for the first criterion are established. Numerical experiments, including tests with highly variable piecewise constant coefficients and a GPU-accelerated three-dimensional elliptic solver, demonstrate that the proposed criteria efficiently avoid both premature termination and over-solving.
Paper Structure (24 sections, 6 theorems, 80 equations, 10 figures, 5 tables)

This paper contains 24 sections, 6 theorems, 80 equations, 10 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Formulation
  3. Error estimation for the conjugate gradient algorithm
  4. Survey of A Posteriori error estimators
  5. Residual estimate
  6. Flux recovery-based estimator
  7. Stopping Criteria derived from the residual
  8. Globally constant coefficient
  9. Highly variable piecewise constant coefficient
  10. Reliability and efficiency of the criterion
  11. Numerical experiments
  12. Test problem 1: isotropic mesh
  13. Test problem 3: highly variable piecewise constant coefficients
  14. Subdomain-based stopping criterion
  15. Test problem 4: revisiting the problem with highly variable coefficients
  16. ...and 9 more sections

Key Result

Theorem 3.2

Let $\Omega\subset \mathbb{R}^2$ be a bounded domain and let $0<\varepsilon, 0<\tau<1/2$. Suppose $u\in H_{0, \Gamma_D}^1(\Omega)$ is the solution to the weak form given in eq:weak, and $f\in L^2(\Omega)$ is the forcing function. Let $u_h\in \mathcal{V}_{h,N}$ satisfy the finite element approximatio then there exists a constant $C(\varepsilon)>0$ depending on $\varepsilon$, but independent of the

Figures (10)

  • Figure 1: Convergence history for test problem 1 (isotropic mesh) with $N=6$. Left: the total error, the ${\bf A}$-norm error $\|{\bf x}_k-{\bf x}\|_{{\bf A}}$, the norm of the linear system residual$\|{\bf r}_k\|$ and $\eta_{\text{RF}}$. Right: the total error, the ${\bf A}$-norm error $\|{\bf x}_k-{\bf x}\|_{{\bf A}}$ and its estimator $\eta_{\text{alg}}$ (delay parameter $d=10$), and the error indicators $\eta_{\text{R}} {, \eta_{\text{MR}},} {\text{ and }} \eta_{\text{FC}} {\text{ and }\eta_{\text{\underline{BDM}}}}$.
  • Figure 1: The largest and smallest eigenvalues of the stiffness matrix ${{\bf A}}$ and the mass matrix ${\bf M}$ on the reference triangle element using Warp & Blend nodes warburton2006explicit. $N_p=(N+1)(N+2)/2$.
  • Figure 2: Sensitivity of the stopping criteria quality ratios with respect to $\tau$ for test problem 1.
  • Figure 3: Geometry of the domain $\Omega$ in test problem 3.
  • Figure 4: Mesh with 3733 elements for \ref{['4-3-1']} in test problem 3.
  • ...and 5 more figures

Theorems & Definitions (12)

  • Theorem 3.2
  • Proof 1
  • Theorem 3.3
  • Proof 2
  • Example 4.2.1
  • Example 4.2.2
  • Lemma A.1
  • Remark A.2
  • Lemma A.3: Theorem 3.6 in melenk2001residual
  • Lemma A.4: Theorem 4.76 in schwab1998p
  • ...and 2 more