Optimal complexity of goal-oriented adaptive FEM for nonsymmetric linear elliptic PDEs

Abstract

We analyze a goal-oriented adaptive algorithm that aims to efficiently compute the quantity of interest $G(u^\star)$ with a linear goal functional $G$ and the solution $u^\star$ to a general second-order nonsymmetric linear elliptic partial differential equation. The current state of the analysis of iterative algebraic solvers for nonsymmetric systems lacks the contraction property in the norms that are prescribed by the functional analytic setting. This seemingly prevents their application in the optimality analysis of goal-oriented adaptivity. As a remedy, this paper proposes a goal-oriented adaptive iteratively symmetrized finite element method (GOAISFEM). It employs a nested loop with a contractive symmetrization procedure, e.g., the Zarantonello iteration, and a contractive algebraic solver, e.g., an optimal multigrid solver. The various iterative procedures require well-designed stopping criteria such that the adaptive algorithm can effectively steer the local mesh refinement and the computation of the inexact discrete approximations. The main results consist of full linear convergence of the proposed adaptive algorithm and the proof of optimal convergence rates with respect to both degrees of freedom and total computational cost (i.e., optimal complexity). Numerical experiments confirm the theoretical results and investigate the selection of the parameters.

Figures (6)

• Figure 1: Schematic overview of the GOAISFEM algorithm with nested symmetrization and inexact solver.
• Figure 2: Left: Mesh $\mathcal{T}_{15}$ for the problem \ref{['eq:singularity_goal']} generated by Algorithm \ref{['algorithm:afem']} with $\# \mathcal{T}_{15} = 2315$. Right: Mesh $\mathcal{T}_{18}$ for the problem \ref{['eq:geometric_singularity']} with $\# \mathcal{T}_{18} = 2130$, where the Dirichlet boundary part $\Gamma_D$ is marked by red solid lines and the Neumann boundary part $\Gamma_N$ by green dashed lines.
• Figure 3: Convergence history plot of estimator product $\eta_\ell(u_\ell^{{\underline{m}}, {\underline{n}}}) \, \zeta_\ell(z^{\underline{\mu}, \underline{\nu}})$ indicated by bullets and goal error from \ref{['eq:goalError:Estimate']} indicated by diamonds with respect to the cumulative computational work (left) and with respect to the cumulative computational time (right) for the benchmark problem \ref{['eq:singularity_goal']}.
• Figure 4: Convergence history plot of estimator product $\eta_\ell(u_\ell^{{\underline{m}}, {\underline{n}}}) \, \zeta_\ell(z^{\underline{\mu}, \underline{\nu}})$ with respect to the cumulative computational cost (left) and the cumulative computational time (right) for the benchmark problem \ref{['eq:geometric_singularity']}.
• Figure 5: Comparison of cumulative time of the local multigrid solver with the Matlab built-in direct solver mldivide with respect to the cumulative computational cost for the benchmark problem \ref{['eq:geometric_singularity']}.

Theorems & Definitions (24)

• Lemma 1: axioms
• Proposition 2: validity of quasi-orthogonality feischl2022
• Remark 3
• Lemma 4: finite termination of algebraic solver bhimps2023
• Lemma 5: contraction of inexact Zarantonello iteration bhimps2023
• Lemma 6: stability and a posteriori error control
• Lemma 7: case of finite mesh-refinement steps
• proof
• Lemma 8: stability of final iterates
• proof