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Table 2 The \(L^{2}\)-norm error estimates at T = 2

From: Optimal error estimates of the local discontinuous Galerkin methods based on generalized fluxes for 1D linear fifth order partial differential equations

k = 0

θ = 0.8

θ = 1

θ = 1.2

θ = 1.5

 

error

order

error

order

error

order

error

order

N = 10

2.09E−01

1.98E−01

2.08E−01

3.29E−01

N = 20

2.88E−02

1.23

9.16E−02

1.06

9.94E−02

1.07

1.28E−01

1.36

N = 40

3.95E−02

1.17

4.35E−02

1.07

4.92E−02

1.01

6.11E−02

1.07

N = 80

1.84E−02

1.10

2.11E−02

1.04

2.45E−02

1.01

3.08E−02

0.99

k = 1

θ = 0.8

θ = 1

θ = 1.2

θ = 1.5

 

error

order

error

order

error

order

error

order

N = 10

1.62E−02

5.41E−02

8.83E−02

4.58E−02

N = 20

4.33E−03

1.90

1.21E−02

2.16

2.51E−02

1.81

1.34E−02

1.77

N = 40

1.18E−03

1.88

2.81E−03

2.11

7.32E−03

1.78

4.17E−03

1.70

N = 80

3.03E−04

1.96

7.03E−04

2.00

1.93E−03

1.92

1.36E−03

1.61

k = 2

θ = 0.8

θ = 1

θ = 1.2

θ = 1.5

 

error

order

error

order

error

order

error

order

N = 10

3.57E−03

4.90E−03

5.25E−03

6.70E−03

N = 20

4.76E−04

2.91

6.12E−04

3.00

6.97E−04

2.91

8.34E−04

3.01

N = 40

6.57E−05

2.86

7.96E−05

3.01

9.30E−05

2.90

8.96E−05

3.22