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Table 1 Numerical results for \(\pmb{(\beta^{-},\beta^{+})=(1,1\text{,}000)}\) with different PIFE schemes

From: A partially penalty immersed Crouzeix-Raviart finite element method for interface problems

Scheme

Grid

\(\boldsymbol{\|u-u_{h}\|_{0,\Omega }}\)

Order

\(\boldsymbol{|u-u_{h}|_{1,h}}\)

Order

Nonsymmetric PIFE (ε = 1)

8 × 8

4.9337E − 3

-

8.0908E − 2

-

16 × 16

1.1162E − 3

2.1441

4.3830E − 2

0.8844

32 × 32

2.9009E − 4

1.9440

2.2917E − 2

0.9355

64 × 64

6.5676E − 5

2.1431

1.1502E − 2

0.9946

128 × 128

1.8919E − 5

1.7955

5.7744E − 3

0.9941

256 × 256

4.4585E − 6

2.0852

2.8965E − 3

0.9954

Incomplete PIFE (ε = 0)

8 × 8

5.5911E − 3

-

8.1187E − 2

-

16 × 16

1.3044E − 3

2.0997

4.4172E − 2

0.6087

32 × 32

3.2578E − 4

2.0014

2.3182E − 2

0.9301

64 × 64

6.9110E − 5

2.2369

1.1552E − 2

1.0048

128 × 128

1.8967E − 5

1.8654

5.7761E − 3

1.0000

256 × 256

4.4796E − 6

2.0820

2.8974E − 3

0.9953

Symmetric PIFE (ε = −1)

8 × 8

6.0212E − 3

-

8.1462E − 2

-

16 × 16

1.3636E − 3

2.1426

4.4097E − 2

0.8854

32 × 32

3.3744E − 4

2.0147

2.3284E − 2

0.9214

64 × 64

7.1179E − 5

2.2451

1.1612E − 2

1.0037

128 × 128

1.8993E − 5

1.9060

5.7769E − 3

1.0073

256 × 256

4.4914E − 6

2.0802

2.8981E − 3

0.9952