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Generalized preconditioned conjugate gradients for adaptive FEM with optimal complexity

Paula Hilbert, Ani Miraçi, Dirk Praetorius

TL;DR

This work develops and analyzes contractive, multilevel solvers as preconditioners for AFEM with inexact algebraic solvers. It introduces an hp-robust, non-linear, non-symmetric geometric MG preconditioner for GPCG and a linear-symmetric variant for PCG, proving contraction factors independent of $h$ and $p$ and linear-time application. An additive Schwarz preconditioner and a linearized MG variant are also shown to be optimal within AFEM, with convergence guarantees under hp-robust assumptions. Numerical experiments on model problems corroborate the theoretical results, showing robust performance and optimal complexity in AFEM with these solvers. The framework unifies numerical linear algebra with AFEM analysis, enabling efficient, scalable solvers for adaptive hp-FEM on nonuniform meshes.

Abstract

We consider adaptive finite element methods (AFEMs) with inexact algebraic solver for second-order symmetric linear elliptic diffusion problems. We formulate and analyze a non-linear and non-symmetric geometric multigrid preconditioner for the generalized preconditioned conjugate gradient method (GPCG) used to solve the arising finite element systems. Moreover, a linear and symmetric variant of the geometric multigrid preconditioner that is suitable for the (standard) preconditioned conjugate gradient method (PCG) is provided and analyzed. We show that both preconditioners are optimal in the sense that, first, the resulting algebraic solvers admit a contraction factor that is independent of the local mesh size h and the polynomial degree p, and, second, that they can be applied with linear computational complexity. Related to this, quasi-optimal computational cost of the overall adaptive finite element method is addressed. Numerical experiments underline the theoretical findings.

Generalized preconditioned conjugate gradients for adaptive FEM with optimal complexity

TL;DR

This work develops and analyzes contractive, multilevel solvers as preconditioners for AFEM with inexact algebraic solvers. It introduces an hp-robust, non-linear, non-symmetric geometric MG preconditioner for GPCG and a linear-symmetric variant for PCG, proving contraction factors independent of and and linear-time application. An additive Schwarz preconditioner and a linearized MG variant are also shown to be optimal within AFEM, with convergence guarantees under hp-robust assumptions. Numerical experiments on model problems corroborate the theoretical results, showing robust performance and optimal complexity in AFEM with these solvers. The framework unifies numerical linear algebra with AFEM analysis, enabling efficient, scalable solvers for adaptive hp-FEM on nonuniform meshes.

Abstract

We consider adaptive finite element methods (AFEMs) with inexact algebraic solver for second-order symmetric linear elliptic diffusion problems. We formulate and analyze a non-linear and non-symmetric geometric multigrid preconditioner for the generalized preconditioned conjugate gradient method (GPCG) used to solve the arising finite element systems. Moreover, a linear and symmetric variant of the geometric multigrid preconditioner that is suitable for the (standard) preconditioned conjugate gradient method (PCG) is provided and analyzed. We show that both preconditioners are optimal in the sense that, first, the resulting algebraic solvers admit a contraction factor that is independent of the local mesh size h and the polynomial degree p, and, second, that they can be applied with linear computational complexity. Related to this, quasi-optimal computational cost of the overall adaptive finite element method is addressed. Numerical experiments underline the theoretical findings.
Paper Structure (23 sections, 17 theorems, 141 equations, 8 figures, 2 tables, 5 algorithms)

This paper contains 23 sections, 17 theorems, 141 equations, 8 figures, 2 tables, 5 algorithms.

Table of Contents

  1. Introduction
  2. Outline
  3. Setting
  4. Mesh and space hierarchy
  5. Weak formulation
  6. Generalized preconditioned conjugate gradient method
  7. Symmetric preconditioners
  8. Non-linear and non-symmetric preconditioners
  9. Non-linear and non-symmetric multigrid preconditioner
  10. $\textit{hp}$-robust geometric multigrid method
  11. Induced multigrid preconditioner
  12. Linear and symmetric multigrid preconditioner
  13. Optimal additive Schwarz preconditioner
  14. Adaptive FEM with contractive solver
  15. Numerical experiments
  16. ...and 8 more sections

Key Result

Theorem 1

Let $\mathbf{A}_{\ell} \in \mathbb{R}^{N_{\ell}^p \times N_{\ell}^p}$ be a symmetric and positive definite matrix and let $\mathbf{x}_{\ell}^{\star} \in \mathbb{R}^{N_{\ell}^p}$ be the unique solution of the linear system $\mathbf{A}_{\ell}\mathbf{x}_{\ell}^{\star} = \mathbf{b}_{\ell}$. Then, for an

Figures (8)

  • Figure 1: Adaptively refined meshes for the Poisson problem from Section \ref{['prob: lshape']} with $\# \mathcal{T}_{8} = 2490$ (left) and checkerboard problem from Section \ref{['prob: checkerboard']} with $\# \mathcal{T}_{8}= 1242$ (right) for adaptivity parameters $\theta = 0.5$ and $\lambda = 0.1$.
  • Figure 2: History plot of the contraction factor for the Poisson problem from Section \ref{['subsec: model problems']} with $\#\mathcal{T}_{10} = 9723$ for $p=1$ (left) and $\# \mathcal{T}_{10} = 340$ for $p=4$ (right) and solver stopping criterion $\lVvert u_{10}^{\star}-u_{10}^k\rVvert < 10^{-13}$.
  • Figure 3: History plot of the contraction factor for the Poisson problem from Section \ref{['subsec: model problems']} employing the algebraic solvers PCG+MG and PCG+nsMG until $\lVvert u_{10}^{\star}-u_{10}^k\rVvert < 10^{-13}$ or for a maximum of 100 iterations with $\# \mathcal{T}_{10} = 9723$ for $p=1$, $\# \mathcal{T}_{10} = 1096$ for $p=2$, and $\#\mathcal{T}_{10} = 405$ for $p=3$.
  • Figure 4: Convergence of the error estimator $\eta_{\ell}(u_{\ell}^{\underline{k}})$ for the Poisson problem from Section \ref{['subsec: model problems']} with $\theta = 0.5$ and $\lambda = 0.05$.
  • Figure 5: History plot of the number of solver steps with respect to the degrees of freedom for the Poisson problem from Section \ref{['subsec: model problems']} with $p=1$ (left) and $p=3$ (right).
  • ...and 3 more figures

Theorems & Definitions (36)

  • Theorem 1
  • Theorem 2: Contractive solvers as preconditioners
  • proof : Proof
  • Remark 3
  • Remark 4: Computational complexity of Algorithm \ref{['algo: geometric MG']}
  • Remark 5
  • Lemma 6: Levelwise smoothers functional/matrix representation
  • proof : Proof
  • Lemma 7
  • proof
  • ...and 26 more