Scalable recovery-based adaptation on Cartesian quadtree meshes for advection-diffusion-reaction problems

Figure 1.  Example of refinement for a non-conforming mesh: initial configuration (left); refined mesh (center), where node 5 corresponds to the newly added degree of freedom; mesh used to define the basis function $ \bar{\varphi}_5 $ (right).

Figure 2.  Example of coarsening for a non-conforming mesh: initial configuration (left) where the red nodes highlight the degrees of freedom to be removed; coarsened mesh (right).

Figure 3.  Gradient recovery: patch associated with the internal (left) and the boundary (right) node $ (x_i, y_j) $. The mesh edges used in the recovery procedure are labeled in green.

Figure 4.  Solution recovery: local numbering of DOFs in $ \Omega^{(k)} $ characterizing the space $ \tilde{Q}_h^2 $ (left); sketch of a non-conforming configuration when interested in recovering the solution at DOF 6 (right).

Figure 5.  Test case 1. Marking-based adaptation.

Figure 6.  Test case 1. Metric-based adaptation.

Figure 7.  Test case 1. Convergence analysis of the error associated with the discrete and the recovered solution and gradient, as a function of the mesh size $ h_{\mathrm{min}} $ (left); log-log plot of the norm $ \left\|{u - u_h}\right\|_{L^2(\Omega)} $ as a function of the number of DOFs on a uniform and on a metric-based adapted mesh (right).

Figure 8.  Test case 1. Histogram of the element size computed at the last iteration of the metric-based adaptive procedure (left); trend of the effectivity index as a function of the adaptation step (right).

Figure 9.  Test case 1. Scalability analysis of the different phases of the marking-based adaptation procedure: parallel speedup (left) and stacked barplot of the time percentage associated with the different phases of the algorithm (right) as a function of the number of parallel processes.

Figure 10.  Test case 2 (rectilinear discontinuity). 2D (left) and 3D (right) view of the exact solution.

Figure 11.  Test case 2 (rectilinear discontinuity). Marking-based adaptation.

Figure 12.  Test case 2 (rectilinear discontinuity). Metric-based adaptation.

Figure 13.  Test case 2 (rectilinear discontinuity). Convergence analysis of the error associated with the discrete and the recovered solution and gradient, as a function of the mesh size $ h_{\mathrm{min}} $ (left); log-log plot of the norm $ \left\|{u - u_h}\right\|_{L^2(\Omega)} $ as a function of the number of DOFs on a uniform and on a metric-based adapted mesh (right).

Figure 14.  Test case 2 (rectilinear discontinuity). Histogram of the element size computed at the last iteration of the metric-based adaptive procedure (left); trend of the effectivity index as a function of the adaptation step (right).

Figure 15.  Test case 2 (circular discontinuity). 2D (left) and 3D (right) view of the solution.

Figure 16.  Test case 2 (circular discontinuity). Metric-based adaptation.

Figure 17.  Test case 3. 2D (left) and 3D (right) view of the solution.

Figure 18.  Test case 3. Marking-based adaptation.

Figure 19.  Test case 3. Metric-based adaptation.

Figure 20.  Test case 3. Scalability analysis of the different phases of the metric-based adaptation procedure: parallel speedup (left) and stacked barplot of the time percentage associated with the different phases of the algorithm (right) as a function of the number of parallel processes.

Figure 21.  Test case 3. Adaptation based on [44].