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Table 1 Numerical result with time step \(\pmb{dt=0.05}\)

From: Full-discrete adaptive FEM for quasi-parabolic integro-differential PDE-constrained optimal control problem

 

On uniform mesh

On adaptive mesh

N

50,421

50,421

50,421

22,769

3,549

3,549

S

147,168

147,168

147,168

64,463

9,576

9,576

E

96,768

96,768

96,768

41,715

6,048

6,048

error

\(\|E_{u}\|_{L^{2}(L^{2})}\)

\(\|E_{y}\|_{L^{2}(H^{1})}\)

\(\|E_{p}\|_{L^{2}(H^{1})}\)

\(\|E_{u}\|_{L^{2}(L^{2})}\)

\(\|E_{y}\|_{L^{2}(H^{1})}\)

\(\|E_{p}\|_{L^{2}(H^{1})}\)

8.33e−02

1.28e−02

4.75e−01

8.16e−02

5.06e−02

7.70e−01

error

\(\|E_{u}\|_{L^{2}(L^{2})}\)

\(\|E_{y}\|_{L^{2}(L^{2})}\)

\(\|E_{p}\|_{L^{2}(L^{2})}\)

\(\|E_{u}\|_{L^{2}(L^{2})}\)

\(\|E_{y}\|_{L^{2}(L^{2})}\)

\(\|E_{p}\|_{L^{2}(L^{2})}\)

8.33e−02

2.79e−04

1.01e−01

8.16e−02

2.09e−03

1.14e−01

error

\(\|E_{u}\|_{L^{\infty}(L^{2})}\)

\(\|E_{y}\|_{L^{\infty}(L^{2})}\)

\(\|E_{p}\| _{L^{\infty}(L^{2})}\)

\(\|E_{u}\|_{L^{\infty}(L^{2})}\)

\(\|E_{y}\|_{L^{\infty}(L^{2})}\)

\(\|E_{p}\|_{L^{\infty}(L^{2})}\)

3.54e−02

3.88e−04

2.53e−02

3.53e−02

5.85e−04

3.21e−02