Preliminaries: This page shows
- how the finite element basis functions are constructed on a metric graph (see the Basis page for an additional illustration),
- how the eigenvalues and eigenfunctions of the shifted Kirchhoff–Laplacian $(\kappa^2-\Delta_\Gamma)$ operator are computed on the tadpole graph,
- and how to project onto a fine space-time mesh.

Functionality

Functionality: This page shows

Experiment

Experiment: This page shows
- how to construct an exact solution to the addressed problem for error analysis purposes.

Convergence rates

Convergence in 𝘩: This page shows
- the convergence with respect to the spatial mesh size 𝘩.
Convergence in τ: This page shows
- the convergence with respect to the temporal mesh size τ.
Convergence in 𝑚: This page shows
- the convergence with respect to the order 𝑚 of the rational approximation.
Figure 1 shows the convergence results. For details on how these plots were generated, please refer to the hyperlinks above.

$Figure 1: Comparison of theoretical and observed convergence behavior for the $L_2((0,T);L_2(\Gamma))$-error with respect to $h$, $\tau$, and $m$. The left and center plots display the convergence rates in $h$ and $\tau$, respectively, on a $\text{log}_{10}$–$\text{log}_{10}$ scale, while the right plot shows the exponential decay in $m$ on a semi-$\text{log}_{e}$ scale, with $m$ plotted on a square-root scale. Dashed lines indicate the theoretical rates, and solid lines represent the observed error curves. The legend below each plot shows the value of $\alpha$ along with the corresponding theoretical ('theo'), and observed ('obs') rates for each case.$

Figure 1: Comparison of theoretical and observed convergence behavior for the $L_2((0,T);L_2(\Gamma))$-error with respect to $h$, $\tau$, and $m$. The left and center plots display the convergence rates in $h$ and $\tau$, respectively, on a $\text{log}_{10}$–$\text{log}_{10}$ scale, while the right plot shows the exponential decay in $m$ on a semi-$\text{log}_{e}$ scale, with $m$ plotted on a square-root scale. Dashed lines indicate the theoretical rates, and solid lines represent the observed error curves. The legend below each plot shows the value of $\alpha$ along with the corresponding theoretical (‘theo’), and observed (‘obs’) rates for each case.

Optimal control

$Figure 2: Comparison of theoretical and observed convergence behavior for the $L_2((0,T);L_2(\Gamma))$-error with respect to $h$ on a $\text{log}_{10}$–$\text{log}_{10}$ scale. Dashed lines indicate the theoretical rates, and solid lines represent the observed error curves. The legend below each plot shows the value of $\alpha$ along with the corresponding theoretical ('theo'), and observed ('obs') rates for each case.$

Figure 2: Comparison of theoretical and observed convergence behavior for the $L_2((0,T);L_2(\Gamma))$-error with respect to $h$ on a $\text{log}_{10}$–$\text{log}_{10}$ scale. Dashed lines indicate the theoretical rates, and solid lines represent the observed error curves. The legend below each plot shows the value of $\alpha$ along with the corresponding theoretical (‘theo’), and observed (‘obs’) rates for each case.

Convergence in τ: This page shows
- the convergence with respect to the temporal mesh size τ.

$Figure 3: Comparison of theoretical and observed convergence behavior for the $L_2((0,T);L_2(\Gamma))$-error with respect to $\tau$ on a $\text{log}_{10}$–$\text{log}_{10}$ scale. Dashed lines indicate the theoretical rates, and solid lines represent the observed error curves. The legend below each plot shows the value of $\alpha$ along with the corresponding theoretical ('theo'), and observed ('obs') rates for each case.$

Figure 3: Comparison of theoretical and observed convergence behavior for the $L_2((0,T);L_2(\Gamma))$-error with respect to $\tau$ on a $\text{log}_{10}$–$\text{log}_{10}$ scale. Dashed lines indicate the theoretical rates, and solid lines represent the observed error curves. The legend below each plot shows the value of $\alpha$ along with the corresponding theoretical (‘theo’), and observed (‘obs’) rates for each case.

Convergence in 𝑚: This page shows
- the convergence with respect to the order 𝑚 of the rational approximation.

$Figure 4: Comparison of theoretical and observed convergence behavior for the $L_2((0,T);L_2(\Gamma))$-error with respect to $m$ on a semi-$\text{log}_{e}$ scale, with $m$ plotted on a square-root scale. Dashed lines indicate the theoretical rates, and solid lines represent the observed error curves. The legend below each plot shows the value of $\alpha$ along with the corresponding theoretical ('theo'), and observed ('obs') rates for each case.$

Figure 4: Comparison of theoretical and observed convergence behavior for the $L_2((0,T);L_2(\Gamma))$-error with respect to $m$ on a semi-$\text{log}_{e}$ scale, with $m$ plotted on a square-root scale. Dashed lines indicate the theoretical rates, and solid lines represent the observed error curves. The legend below each plot shows the value of $\alpha$ along with the corresponding theoretical (‘theo’), and observed (‘obs’) rates for each case.

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