1. Fig. 5. Experiment with known solution from Section 8
Fig. 5. Experiment with known solution from Section 8.4: Number of Picard iterations used in each step of Algorithm 4.3 for different values of $\lambda\in\{0.1,\dots, 10^{-6} \}$ and $\theta = 0.2$ (left) as well as $\theta = 0.8$ (right). As expected, we observe logarithmic growth for naive initial guesses $u_\ell^0 := 0$ (dashed lines); see Remark 3.3. On the other hand, nested iteration $u_\ell^0 := u_{\ell-1}$ (solid lines) leads to a bounded number of Picard iterations (see Remark 4.8).
From: Rate optimal adaptive FEM with inexact solver for nonlinear operators
IMA J Numer Anal. Published online September 15, doi: /imanum/drx050
IMA J Numer Anal | © The authors Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.This is an Open Access article distributed under the terms of the Creative Commons Attribution License ( which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited.

2. Fig. 1. Experiment with known solution from Section 8
Fig. 1. Experiment with known solution from Section 8.4: $Z$-shaped domain $\varOmega \subset \mathbb{R}^2$ and the initial mesh $\mathcal{T}_0$ (left), and with NVB adaptively generated mesh $\mathcal{T}_{19}$ with $5854$ elements (right). $\Gamma_D$ is visualized by a thick red line.
From: Rate optimal adaptive FEM with inexact solver for nonlinear operators
IMA J Numer Anal. Published online September 15, doi: /imanum/drx050
IMA J Numer Anal | © The authors Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.This is an Open Access article distributed under the terms of the Creative Commons Attribution License ( which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited.

3. Fig. 2. Experiment with known solution from Section 8
Fig. 2. Experiment with known solution from Section 8.4: Convergence of $\eta_\ell(u_\ell)$ (solid lines) and $\left\| {{\nabla u^\star - \nabla u_\ell}} \right\|_{L^2(\varOmega)}$ (dashed lines) for $\lambda = 0.1$ and different values of $\theta\in\{0.2,\dots, 1\}$, where we compare naive initial guesses $u_\ell^0 := 0$ (left) and nested iteration $u_\ell^0 := u_{\ell-1}$ (right).
From: Rate optimal adaptive FEM with inexact solver for nonlinear operators
IMA J Numer Anal. Published online September 15, doi: /imanum/drx050
IMA J Numer Anal | © The authors Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.This is an Open Access article distributed under the terms of the Creative Commons Attribution License ( which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited.

4. Fig. 3. Experiment with known solution from Section 8
Fig. 3. Experiment with known solution from Section 8.4: Convergence of $\eta_\ell(u_\ell)$ (solid lines) and $\left\| {{\nabla u^\star - \nabla u_\ell}} \right\|_{L^2(\varOmega)}$ (dashed lines) for $\lambda = 10^{-5}$ and different values of $\theta\in\{0.2,\dots, 1 \}$, where we compare naive initial guesses $u_\ell^0 := 0$ (left) and nested iteration $u_\ell^0 := u_{\ell-1}$ (right).
From: Rate optimal adaptive FEM with inexact solver for nonlinear operators
IMA J Numer Anal. Published online September 15, doi: /imanum/drx050
IMA J Numer Anal | © The authors Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.This is an Open Access article distributed under the terms of the Creative Commons Attribution License ( which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited.

5. Fig. 4. Experiment with known solution from Section 8
Fig. 4. Experiment with known solution from Section 8.4: Convergence of $\eta_\ell(u_\ell)$ (solid lines) and $\left\| {{\nabla u^\star - \nabla u_\ell}} \right\|_{L^2(\varOmega)}$ (dashed lines) for $\theta = 0.2$, and different values of $\lambda\in\{1,0.1, 0.01,\dots, 10^{-6} \}$, where we compare naive initial guesses $u_\ell^0 := 0$ (left) and nested iteration $u_\ell^0 := u_{\ell-1}$ (right).
From: Rate optimal adaptive FEM with inexact solver for nonlinear operators
IMA J Numer Anal. Published online September 15, doi: /imanum/drx050
IMA J Numer Anal | © The authors Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.This is an Open Access article distributed under the terms of the Creative Commons Attribution License ( which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited.

6. Fig. 6. Experiment with unknown solution from Section 8
Fig. 6. Experiment with unknown solution from Section 8.5: Convergence of $\eta_\ell(u_\ell)$ for $\lambda = 0.1$ and $\theta\in\{0.2,\dots, 1\}$, where we compare naive initial guesses $u_\ell^0 := 0$ (left) and nested iteration $u_\ell^0 := u_{\ell-1}$ (right).
From: Rate optimal adaptive FEM with inexact solver for nonlinear operators
IMA J Numer Anal. Published online September 15, doi: /imanum/drx050
IMA J Numer Anal | © The authors Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.This is an Open Access article distributed under the terms of the Creative Commons Attribution License ( which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited.

7. Fig. 7. Experiment with unknown solution from Section 8
Fig. 7. Experiment with unknown solution from Section 8.5: Convergence of $\eta_\ell(u_\ell)$ for $\lambda = 10^{-5}$ and $\theta\in\{0.2,\dots, 1\}$, where we compare naive initial guesses $u_\ell^0 := 0$ (left) and nested iteration $u_\ell^0 := u_{\ell-1}$ (right).
From: Rate optimal adaptive FEM with inexact solver for nonlinear operators
IMA J Numer Anal. Published online September 15, doi: /imanum/drx050
IMA J Numer Anal | © The authors Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.This is an Open Access article distributed under the terms of the Creative Commons Attribution License ( which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited.

8. Fig. 8. Experiment with unknown solution from Section 8
Fig. 8. Experiment with unknown solution from Section 8.5: Convergence of $\eta_\ell(u_\ell)$ for $\theta = 0.2$ and $\lambda\in\{1,0.1, 0.01,\dots, 10^{-5} \}$, where we compare naive initial guesses $u_\ell^0 := 0$ (left) and nested iteration $u_\ell^0 := u_{\ell-1}$ (right).
From: Rate optimal adaptive FEM with inexact solver for nonlinear operators
IMA J Numer Anal. Published online September 15, doi: /imanum/drx050
IMA J Numer Anal | © The authors Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.This is an Open Access article distributed under the terms of the Creative Commons Attribution License ( which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited.

9. Fig. 9. Experiment with unknown solution from Section 8
Fig. 9. Experiment with unknown solution from Section 8.5: Number of Picard iterations used in each step of Algorithm 4.3 for different values of $\lambda\in\{1, 0.1,\dots, 10^{-5} \}$ and $\theta = 0.2$ (left) as well as $\theta = 0.8$ (right). As expected, we observe logarithmic growth for naive initial guesses $u_\ell^0 := 0$ (dashed lines) as well as a bounded number of Picard iterations for nested iteration $u_\ell^0 := u_{\ell-1}$ (solid lines).
From: Rate optimal adaptive FEM with inexact solver for nonlinear operators
IMA J Numer Anal. Published online September 15, doi: /imanum/drx050
IMA J Numer Anal | © The authors Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.This is an Open Access article distributed under the terms of the Creative Commons Attribution License ( which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited.