Goal-oriented adaptive finite element methods with optimal computational complexity

TL;DR

This work develops GOAFEM for a linear symmetric elliptic PDE with a linear goal functional, integrating goal-oriented error control with mesh refinement and inexact, contractive solvers. The authors establish linear convergence of the estimator-product and prove that, for sufficiently small adaptivity parameters, the method attains optimal rates with respect to both the number of elements and the total computational cost, effectively matching convergence with respect to DOFs to convergence with respect to runtime. The analysis hinges on robust a posteriori estimators for the primal and dual problems, axioms of adaptivity, and a contractive solver framework (e.g., PCG with optimal Schwarz preconditioners or geometric multigrid). Numerical experiments confirm the theoretical results across singular and geometrically challenging cases, highlighting the importance of (i) inexact solving and (ii) effective marking strategies in achieving cost-effective goal accuracy.

Abstract

We consider a linear symmetric and elliptic PDE and a linear goal functional. We design and analyze a goal-oriented adaptive finite element method, which steers the adaptive mesh-refinement as well as the approximate solution of the arising linear systems by means of a contractive iterative solver like the optimally preconditioned conjugate gradient method or geometric multigrid. We prove linear convergence of the proposed adaptive algorithm with optimal algebraic rates. Unlike prior work, we do not only consider rates with respect to the number of degrees of freedom but even prove optimal complexity, i.e., optimal convergence rates with respect to the total computational cost.

Figures (5)

• Figure 1: Left: Initial mesh $\mathcal{T}_0$. The shaded area is the set $T_1$ from Section \ref{['subsec:example1']}. Right: Mesh after $14$ iterations of Algorithm \ref{['algorithm']} with $\#\mathcal{T}_{14} = 4157$.
• Figure 2: Comparison between iterative solvers for the problem from Section \ref{['subsec:example1']}. A conjugate gradient method without preconditioner (CG) leads to optimal rates with respect to $\#\mathcal{T}_\ell$ for the final iterates where $k = {\underline{k}}(\ell)$, but not with respect to ${\tt work}(\ell,k)$ for every $(\ell, k) \in \mathcal{Q}$. Our choice of the iterative solver (ML) achieves optimal rates with respect to both measures.
• Figure 3: Comparison between $\Xi_\ell^k$, discrete goal $G_\ell(u_\ell^k, z_\ell^k)$, primal residual evaluated at the dual solution $z_\ell^k$, and direct evaluation of goal functional $G(u_\ell^k)$ for every iterate $(\ell,k) \in \mathcal{Q}$ and different values of $\lambda_{\rm ctr} \in \{ 1, 10^{-2}, 10^{-4}, 10^{-6} \}$. The primal residual evaluated at the dual solution $z_\ell^k$ is the difference between goal and discrete goal; see \ref{['eq:def:goal']}.
• Figure 4: Left: Initial mesh $\mathcal{T}_0$. The shaded area is the set $T_2$ from Section \ref{['subsec:example2']} and the Dirichlet boundary at the re-entrant corner is marked in red. Right: Mesh after $13$ iterations of Algorithm \ref{['algorithm']} with $\#\mathcal{T}_{13} = 4534$.
• Figure 5: Rates of the estimator product for final iterates over $\#\mathcal{T}_\ell$ and $\Xi_\ell^k$ as well as goal error over ${\tt work}(\ell,k)$ for all $(\ell,k) \in \mathcal{Q}$.

Theorems & Definitions (11)

• proof
• proof : Proof of \ref{['eq:lctr-prime']}
• proof : Proof of Lemma \ref{['lemma:banach2-new']}(i)
• proof : Proof of Lemma \ref{['lemma:banach2-new']}(ii)
• proof : Proof of Lemma \ref{['lemma:banach2-new']}(iii)
• proof : Proof of Lemma \ref{['lemma:banach2-new']}(iv)
• proof : Proof of Lemma \ref{['lemma:banach2-new']} (choosing $\mu$)
• proof : Proof of Lemma \ref{['lemma:banach2-new']}(v)
• proof : Proof of Theorem \ref{['theorem:linear-convergence']}
• proof