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Table 11 Convergence orders of \(\pmb{\mathcal {O}(h^{\mu})}\) of the decoupled Algorithm  3.4 at time \(\pmb{T=1.0}\) , with varying mesh h but fixed time step \(\pmb{\Delta t=0.01}\)

From: Stability and convergence of some novel decoupled schemes for the non-stationary Stokes-Darcy model

1/ h \(\boldsymbol {\|{\mathbf{u}}_{3.4}^{m,h}-{\mathbf{u}}_{3.4}^{m,\frac{h}{2}}\|_{0}}\) \(\boldsymbol {\rho_{{\mathbf{u}}_{f},h,0}}\) \(\boldsymbol {\|{\mathbf{u}}_{3.4}^{m,h}-{\mathbf{u}}_{3.4}^{m,\frac{h}{2}}\|_{1}}\) \(\boldsymbol {\rho_{{\mathbf{u}}_{f},h,1}}\) \(\boldsymbol {\|p_{3.4}^{m,h}-p_{3.4}^{m,\frac{h}{2}}\|_{0}}\) \(\boldsymbol {\rho_{p_{f},h,0}}\)
2 0.215106 3.80487 1.65033 1.91101 0.941814 1.50623
4 0.0565345 3.86891 0.86359 1.93145 0.625278 2.43518
8 0.0146125 4.04563 0.447119 2.13703 0.256769 2.85676
16 0.00361192   0.209225   0.0898811  
1/ h \(\boldsymbol {\|\phi_{3.4}^{m,h}-\phi_{3.4}^{m,\frac{h}{2}}\|_{0}}\) \(\boldsymbol {\rho_{\phi,h,0}}\) \(\boldsymbol {\|\phi_{3.4}^{m,h}-\phi_{3.4}^{m,\frac{h}{2}}\|_{1}}\) \(\boldsymbol {\rho_{\phi,h,1}}\)
2 0.135175 3.3141 1.30791 1.68817
4 0.0407879 4.07829 0.774753 1.90637
8 0.0100012 4.18768 0.406401 1.98302
16 0.00238826   0.204943