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# Comparison of the error in the mean drag for the Stokes problem for uniform and adaptive spatio-temporal refinement

Source publication

In this work, the dual-weighted residual method is applied to a space-time formulation of nonstationary Stokes and Navier-Stokes flow. Tensor-product space-time finite elements are being used to discretize the variational formulation with discontinuous Galerkin finite elements in time and inf-sup stable Taylor-Hood finite element pairs in space. To...

## Contexts in source publication

**Context 1**

... it can be seen that the effectivity index converges towards 1 for uniform spatial refinement, which proves numerically that in the limit our PU-DWR error estimator and the true error coincide. Looking at the rows of Figure 3, i.e. keeping the number of spatial degrees of freedom fixed and only refining uniformly in time, we see that the spatial error remains almost constant. This verifies that our spatial error does not depend on temporal refinement. ...

**Context 2**

... ref. + H 1 0 proj. In Figure 3, we start with a common coarse spatio-temporal mesh and iteratively refine uniformly in space and time or adaptively refine in space and time based on Algorithm 2. As expected the dual weighted residual method driven mesh refinement leads to a faster convergence in the error in the mean drag functional. In particular, for adaptive refinement a lot fewer degrees of freedom are needed to reach a prescribed accuracy in the goal functional, e.g. to achieve a tolerance of 2 · 10 −3 using uniform refinement requires 7,590,400 space-time degrees of freedom, whereas adaptive refinement only needs 902,838 space-time degrees of freedom for the primal problem. ...

**Context 3**

... Figure 10, we visualize the true error in the mean squared vorticity for the backward facing step for uniform and adaptive spatio-temporal refinement. Analogous to Figure 3 and Figure 5 for the (Navier-)Stokes 2D-3 benchmark, we detect that the error of our adaptively refined solution converges with a higher rate than a uniformly refined solution. In particular, when comparing the fourth refinement cycle, we observe that adaptive refinement uses only 1 6 of the number of space-time degrees of freedom as uniform refinement to reach a tolerance of 0.7 in the goal functional. ...

**Context 4**

... Figure 12, we plot the snapshot of the vorticity magnitude of the primal backward-facing step problem at time t ≈ 4. It can be seen that the vorticity magnitude is largest at the reentrant corner x = (0, 1 2 ). In Figure 13, we display a 4 times adaptively refined grid with divergence-free L 2 projection. On the one hand, we observe that the grid is mostly being refined close to the reentrant corner where the vorticity magnitude is at its maximum. ...

**Context 5**

... it can be seen that the effectivity index converges towards 1 for uniform spatial refinement, which proves numerically that in the limit our PU-DWR error estimator and the true error coincide. Looking at the rows of Figure 3, i.e. keeping the number of spatial degrees of freedom fixed and only refining uniformly in time, we see that the spatial error remains almost constant. This verifies that our spatial error does not depend on temporal refinement. ...

**Context 6**

... ref. + H 1 0 proj. In Figure 3, we start with a common coarse spatio-temporal mesh and iteratively refine uniformly in space and time or adaptively refine in space and time based on Algorithm 2. As expected the dual weighted residual method driven mesh refinement leads to a faster convergence in the error in the mean drag functional. In particular, for adaptive refinement a lot fewer degrees of freedom are needed to reach a prescribed accuracy in the goal functional, e.g. to achieve a tolerance of 2 · 10 −3 using uniform refinement requires 7,590,400 space-time degrees of freedom, whereas adaptive refinement only needs 902,838 space-time degrees of freedom for the primal problem. ...

**Context 7**

... Figure 10, we visualize the true error in the mean squared vorticity for the backward facing step for uniform and adaptive spatio-temporal refinement. Analogous to Figure 3 and Figure 5 for the (Navier-)Stokes 2D-3 benchmark, we detect that the error of our adaptively refined solution converges with a higher rate than a uniformly refined solution. In particular, when comparing the fourth refinement cycle, we observe that adaptive refinement uses only 1 6 of the number of space-time degrees of freedom as uniform refinement to reach a tolerance of 0.7 in the goal functional. ...

**Context 8**

... Figure 12, we plot the snapshot of the vorticity magnitude of the primal backward-facing step problem at time t ≈ 4. It can be seen that the vorticity magnitude is largest at the reentrant corner x = (0, 1 2 ). In Figure 13, we display a 4 times adaptively refined grid with divergence-free L 2 projection. On the one hand, we observe that the grid is mostly being refined close to the reentrant corner where the vorticity magnitude is at its maximum. ...