A Posteriori Error Estimates for Approximate Solutions of Linear Parabolic Problems

Functional A Posteriori Error Estimates for Time-Periodic Parabolic Optimal Control Problems

“…We consider the multiharmonic finite element (MhFE) approximations of the reduced optimality system, and derive guaranteed and fully computable bounds for the discretization errors. For this purpose, we use the functional a posteriori error estimation techniques earlier introduced by S. Repin, see, e.g., the papers on parabolic problems [32,13] as well as on optimal control problems [11,12], the books [33,29] and the references therein. In particular, our functional a posteriori error analysis uses the techniques close to those suggested in [32], but the analysis contains essential changes due to the MhFEM setting.…”

Fully reliable error control for first-order evolutionary problems

“…, which are not included in (12) and contribute to the irremovable gap between the error and the estimate.…”

“…1.90 · 10 −4 2.87 · 10 −4 1.23 Q (6) 3.20 · 10 −4 4.68 · 10 −4 1.21 Q (8) 4.26 · 10 −4 6.21 · 10 −4 1.21 Q (10) 5.02 · 10 −4 7.37 · 10 −4 1.21 Q (12) 5.49 · 10 −4 8.17 · 10 −4 1.22 Q (14) 5.76 · 10 −4 8.69 · 10 −4 1.23 rapidly oscillating on Q, produces approximations changing in time that are depicted in Figure 16. Here, the approximate solution is reproduced on the fixed mesh T h (411 ND) and with K = 15 time steps.…”

“…3.45 · 10 −6 4.92 · 10 −5 3.77 Q (2) 2.68 · 10 −5 1.58 · 10 −4 2.43 Q (4) 7.36 · 10 −5 3.41 · 10 −4 2.15 Q (6) 1.21 · 10 −4 5.45 · 10 −4 2.12 Q (8) 1.57 · 10 −4 6.44 · 10 −4 2.02 Q (10) 1.84 · 10 −4 6.98 · 10 −4 1.95 Q (12) 2.02 · 10 −4 7.39 · 10 −4 1.91 Q (14) 2.18 · 10 −4 7.77 · 10 −4 1.89 Table 3: Ex. 5.…”

“…4. Initial mesh (a) and the meshes produced during the refinement on Q(9) and Q(12) time-slabs based either on the true error (b), (d) or on the majorant (c), (e), using the bulk marking M 0.1 .…”