A posteriori error estimates based on multilevel decompositions with large problems on the coarsest level

TL;DR

The work addresses residual-based a posteriori error estimation in multilevel discretizations when the coarsest-level system is large and only approximately solved. It introduces a novel adaptive strategy using conjugate gradient iterations to approximate the coarsest-term $\mathbf{r}_0^T \mathbf{A}_0^{-1} \mathbf{r}_0$ while preserving efficiency and robustness with respect to the coarsest problem size. By comparing existing estimators (Becker, Ruede, Harbrecht) and deriving a new stable-splitting-based bound, the paper provides computable, scalable error measures validated on a 3D Poisson problem; numerical experiments show robustness to the number of levels and coarse-size variations. The results offer practical guidance for implementing reliable multilevel error estimators in large-scale and parallel PDE solvers, with potential extensions to inexact arithmetic.

Abstract

Multilevel methods represent a powerful approach in numerical solution of partial differential equations. The multilevel structure can also be used to construct estimates for total and algebraic errors of computed approximations. This paper deals with residual-based error estimates that are based on properties of quasi-interpolation operators, stable-splittings, or frames. We focus on the settings where the system matrix on the coarsest level is still large and the associated terms in the estimates can only be approximated. We show that the way in which the error term associated with the coarsest level is approximated is substantial. It can significantly affect both the efficiency (accuracy) of the overall error estimates and their robustness with respect to the size of the coarsest problem. The newly proposed approximation of the coarsest-level term is based on using the conjugate gradient method with an appropriate stopping criterion. We prove that the resulting estimates are efficient and robust with respect to the size of the coarsest-level problem. Numerical experiments illustrate the theoretical findings.

Figures (3)

• Figure 1: Efficiency indices $I_1$ (\ref{['plot:rude']}) and $I_2$ (\ref{['plot:BJR']}), \ref{['eq:numexp2_effb']} and \ref{['eq:numexp2_effa']}, for varying number of levels $J$. We plot the efficiency for approximations $v_J$ and for the associated intermediate results after each V-cycle; each corresponds to a single mark.
• Figure 2: Efficiency indices $I_3$\ref{['eq:numexp1_eff']} for the experiment in \ref{['sec:numexp_coarse_size']}. The estimates differ in the way of approximating the coarsest-level term $\|\nabla r_0 \|^2 = \mathbf{r}^*_0\mathbf{A}^{-1}_0\mathbf{r}_0$. This term is: computed by a direct solver for the coarsest problem (i), approximated using four iterations of the CG solver (ii), approximated by replacing the stiffness matrix by its scaled diagonal approximation (iii), determined using the adaptive CG approximation (iv).
• Figure 3: Number of CG iterations determined by the adaptive approach described in \ref{['sec:strategy']}, which is used to estimate the residual norm $\|\nabla r_0 \|$ associated with the coarsest level. The horizontal axis indicates the number of V-cycles used in computing the approximation $v_J$.

Theorems & Definitions (45)

• Lemma 1: Bramble--Hilbert lemma
• Lemma 2: Friedrich's inequality
• Lemma 3: Trace inequality
• Lemma 4: Inverse inequality
• Lemma 5
• proof
• Theorem 1
• proof
• Theorem 2
• Theorem 3