Iterative solvers in adaptive FEM

TL;DR

This chapter surveys iterative solvers embedded in adaptive finite element methods for second-order elliptic PDEs, presenting a unified framework that couples local mesh refinement with contractive solvers to achieve rate-optimal convergence at minimal computational cost. It centers on residual-based a-posteriori estimators and a quasi-error formulation, enabling full R-linear convergence and, under mild adaptivity parameter choices, optimal complexity for symmetric problems. The text extends these results to goal-oriented adaptivity, non-symmetric problems via symmetrization, and nonlinear problems via energy-based iterates, supported by numerical experiments that demonstrate practical efficiency and robustness. The findings provide a principled pathway to implement cost-effective AFEMs with provable convergence properties across a broad class of PDEs and solver paradigms.

Abstract

This chapter provides an overview of state-of-the-art adaptive finite element methods (AFEMs) for the numerical solution of second-order elliptic partial differential equations (PDEs), where the primary focus is on the optimal interplay of local mesh refinement and iterative solution of the arising discrete systems. Particular emphasis is placed on the thorough description of the essential ingredients necessary to design adaptive algorithms of optimal complexity, i.e., algorithms that mathematically guarantee the optimal rate of convergence with respect to the overall computational cost and, hence, time. Crucially, adaptivity induces reliability of the computed numerical approximations by means of a-posteriori error control. This ensures that the error committed by the numerical scheme is bounded from above by computable quantities. The analysis of the adaptive algorithms is based on the study of appropriate quasi-error quantities that include and balance different components of the overall error. Importantly, the quasi-errors stemming from an adaptive algorithm with contractive iterative solver satisfy a centerpiece concept, namely, full R-linear convergence. This guarantees that the adaptive algorithm is essentially contracting this quasi-error at each step and it turns out to be the cornerstone for the optimal complexity of AFEM. The unified analysis of the adaptive algorithms is presented in the context of symmetric linear PDEs. Extensions to goal-oriented, non-symmetric, as well as non-linear PDEs are presented with suitable nested iterative solvers fitting into the general analytical framework of the linear symmetric case. Numerical experiments highlight the theoretical results and emphasize the practical relevance and gain of adaptivity with iterative solvers for numerical simulations with optimal complexity.

Figures (19)

• Figure 1: Visualization in 2D of the finite number of patterns of new triangles obtained by NVB. Each triangle has an associated refinement edge to be bisected in case the triangle is marked for refinement. NVB marks edges, here indicated by red dots, in order to avoid hanging nodes. The red line indicates the new refinement edge -- opposite to the newest vertex -- on the new triangles.
• Figure 2: Example in 2D of the overlay of two distinct triangulations $\mathcal{T}_h$ and $\mathcal{T}_{h'}$, represented in red and blue. The meshes stem from the refinement of the same coarse mesh $\mathcal{T}_H$, represented in black, and their overlay $\mathcal{T}_h \oplus \mathcal{T}_{h'}$, i.e., the coarsest common refinement, is illustrated in the rightmost figure.
• Figure 3: Example in 2D of non-locality of the mesh closure. From the initial triangulation $\mathcal{T}_0$ in the left, iterative NVB refinement leads to the middle triangulation $\mathcal{T}_\ell$. Only four elements (highlighted in gray, each refinement edge represented by a red line) of $\mathcal{T}_\ell$ are marked for refinement. Nonetheless, each triangle of $\mathcal{T}_\ell$ is bisected at least once for the new triangulation $\mathcal{T}_{\ell+1}$ in the right to be conforming (and avoid hanging nodes) and thus the refinement propagates through the entire mesh.
• Figure 4: Initial mesh, adaptively refined mesh, exact solution $u^\star$, and discrete solution $u_\ell^{\underline{k}}$ for the Kellogg benchmark problem from Subsection \ref{['BMP:sec:numerics']}. The results are generated by Algorithm \ref{['BMP:algorithm:afem']} with polynomial degree $p = 2$, bulk parameter $\theta = 0.5$, and stopping parameter $\lambda_{\textup{alg}} = 0.01$.
• Figure 5: Convergence plot of Algorithm \ref{['BMP:algorithm:afem']} to solve the Kellogg benchmark problem from Subsection \ref{['BMP:sec:numerics']} for various polynomial degrees. The adaptivity parameters read $\theta = 0.5$ and $\lambda_{\textup{alg}} = 0.01$. All graphs display values of the residual-based error estimator $\eta_\ell (u_\ell^{{\underline{k}}})$ from \ref{['BMP:eq:estimator']}.

Theorems & Definitions (36)

• Lemma 1: perturbed reduction of the estimator for nested approximations
• proof
• Remark 2
• Lemma 3: a-posteriori control of the overall error
• proof
• Lemma 4: contraction of quasi-error of final iterates
• proof
• Theorem 5: parameter-robust full R-linear convergence
• Theorem 6: optimal complexity
• Lemma 7: tail summability vs. R-linear convergence