1 Introduction
2 Statement and Solvability of the Continuous Problem
2.1 Oseen Problem in Terms of Velocity–Vorticity–Pressure
2.2 Defining a Weak Formulation
2.3 Solvability Analysis
3 Finite Element Discretisation
3.1 Defining the Discrete Problem
3.2 Solvability and Stability of the Discrete Problem
3.3 A Priori Error Estimates
4 Discontinuous Galerkin Method
4.1 Preliminaries
4.2 Discrete Formulation and Solvability Analysis
4.3 A Priori Error Bounds
5 Numerical Tests
5.1 Test 1: Experimental Convergence in 2D
DoF |
\(\Vert \varvec{u}-\varvec{u}_h\Vert _{\mathrm {H}}\)
| rate
|
\(\Vert \varvec{\omega }-\varvec{\omega }_h\Vert _{\mathrm {Z}}\)
| rate
|
\(\Vert p-p_h\Vert _{0,\varOmega }\)
| rate
|
\(\Vert \mathop {\mathrm {div}}\nolimits \varvec{u}_h\Vert _{\ell ^\infty }\)
|
---|---|---|---|---|---|---|---|
\(k=0\)
| |||||||
34 | 0.1357 | – | 1.2943 | – | 0.2002 | – | 1.0e−14 |
114 | 0.1129 | 0.2654 | 1.0072 | 0.3612 | 0.1219 | 0.7161 | 1.3e−14 |
418 | 0.0619 | 0.8655 | 0.5623 | 0.8405 | 0.0572 | 1.0910 | 1.3e−15 |
1602 | 0.0315 | 0.9763 | 0.2869 | 0.9707 | 0.0280 | 1.0311 | 1.2e−15 |
6274 | 0.0158 | 0.9952 | 0.1441 | 0.9937 | 0.0139 | 1.0072 | 1.0e−14 |
24,834 | 0.0079 | 0.9989 | 0.0721 | 0.9985 | 0.0069 | 1.0022 | 1.3e−15 |
98,818 | 0.0039 | 0.9997 | 0.0361 | 0.9996 | 0.0035 | 1.0000 | 1.7e−13 |
\(k=1\)
| |||||||
98 | 0.0980 | – | 0.8753 | – | 0.0624 | – | 2.4e−15 |
354 | 0.0337 | 1.5384 | 0.3448 | 1.3441 | 0.0173 | 1.8502 | 4.3e−14 |
1346 | 0.0094 | 1.8472 | 0.0979 | 1.8173 | 0.0038 | 2.1804 | 3.8e−13 |
5250 | 0.0024 | 1.9514 | 0.0255 | 1.9403 | 8.3e−04 | 2.1952 | 8.0e−14 |
20,738 | 6.4e−04 | 1.9873 | 0.0064 | 1.9835 | 1.9e−04 | 2.0791 | 1.2e−15 |
82,434 | 2.2e−04 | 1.9973 | 0.0016 | 1.9960 | 4.8e−05 | 2.0233 | 6.2e−15 |
328,706 | 3.8e−05 | 1.9992 | 4.1e−04 | 1.9992 | 1.2e−05 | 2.0064 | 1.4e−14 |
\(k=2\)
| |||||||
194 | 0.0563 | – | 0.5138 | – | 0.0237 | – | 1.7e−13 |
722 | 0.0078 | 2.8844 | 0.0893 | 2.7524 | 0.0024 | 3.2137 | 5.8e−14 |
2786 | 0.0011 | 2.9302 | 0.0121 | 2.9886 | 1.8e−04 | 3.1729 | 4.6e−14 |
10,946 | 1.3e−04 | 2.9872 | 0.0015 | 2.9873 | 1.4e−05 | 3.2661 | 4.3e−15 |
43,394 | 1.6e−05 | 2.9992 | 1.9e−04 | 3.0002 | 1.4e−06 | 3.2349 | 2.7e−15 |
172,802 | 2.1e−06 | 2.9981 | 2.3e−05 | 3.0014 | 1.6e−07 | 3.1252 | 4.9e−15 |
689,666 | 5.3e−07 | 2.9840 | 8.2e−06 | 3.0070 | 3.7e−08 | 2.9206 | 5.0e−14 |
DoF |
h
|
\(\Vert \varvec{u}-\varvec{u}_h\Vert _{\mathrm {H}}\)
| rate
|
\(\Vert \varvec{\omega }-\varvec{\omega }_h\Vert _{\mathrm {Z}}\)
| rate
|
\(\Vert p-p_h\Vert _{0,\varOmega }\)
| rate
|
\(\Vert \mathop {\mathrm {div}}\nolimits \varvec{u}_h\Vert _{\ell ^\infty }\)
|
---|---|---|---|---|---|---|---|---|
\(\nu =0.01\)
| ||||||||
34 | 0.7071 | 0.1327 | – | 0.1709 | – | 191.8 | – | 1.76e−14 |
114 | 0.3536 | 0.1108 | 0.2601 | 0.1241 | 0.4615 | 107.3 | 0.8384 | 1.71e−13 |
418 | 0.1768 | 0.0615 | 0.8605 | 0.0610 | 1.0324 | 55.17 | 0.9597 | 5.22e−12 |
1602 | 0.0884 | 0.0313 | 0.9614 | 0.0293 | 1.0458 | 27.78 | 0.9943 | 4.23e−10 |
6274 | 0.0442 | 0.0157 | 0.9901 | 0.0144 | 1.0157 | 13.91 | 0.9975 | 3.84e−09 |
24834 | 0.0221 | 0.0079 | 0.9975 | 0.0072 | 1.0040 | 6.9591 | 0.9994 | 3.28e−08 |
98818 | 0.0111 | 0.0039 | 0.9994 | 0.0036 | 1.0012 | 3.4833 | 0.9998 | 6.46e−07 |
\(\nu =0.0001\)
| ||||||||
34 | 0.7071 | 0.1327 | – | 0.01188 | – | 191.8 | – | 2.31e−14 |
114 | 0.3536 | 0.1107 | 0.2617 | 0.0061 | 0.9457 | 107.3 | 0.8384 | 1.65e−13 |
418 | 0.1768 | 0.0613 | 0.8599 | 0.0012 | 2.1252 | 55.17 | 0.9597 | 3.11e−12 |
1602 | 0.0884 | 0.0313 | 0.9606 | 0.0003 | 1.3765 | 27.78 | 0.9944 | 2.73e−10 |
6274 | 0.0442 | 0.0158 | 0.9899 | 0.0001 | 1.2588 | 13.91 | 0.9975 | 7.63e−09 |
24834 | 0.0221 | 0.0079 | 0.9974 | 7.36e−05 | 1.0832 | 6.9590 | 0.9994 | 1.64e−08 |
98818 | 0.0111 | 0.0039 | 0.9994 | 3.62e−05 | 1.0223 | 3.4833 | 0.9998 | 1.89e−07 |
\(\nu =0.000001\)
| ||||||||
34 | 0.7071 | 0.1327 | – | 0.0011 | – | 191.8 | – | 2.73e−14 |
114 | 0.3536 | 0.1107 | 0.2617 | 0.0006 | 0.9621 | 107.3 | 0.8384 | 1.96e−13 |
418 | 0.1768 | 0.0613 | 0.8599 | 0.0001 | 1.4123 | 55.17 | 0.9597 | 3.98e−12 |
1602 | 0.0884 | 0.0313 | 0.9606 | 2.42e−05 | 1.2313 | 27.78 | 0.9932 | 1.39e−10 |
6274 | 0.0442 | 0.0157 | 0.9899 | 9.81e−06 | 1.0495 | 13.91 | 0.9975 | 1.06e−08 |
24834 | 0.0221 | 0.0079 | 0.9974 | 1.56e−06 | 1.5904 | 6.9591 | 0.9994 | 6.15e−07 |
98818 | 0.0111 | 0.0039 | 0.9994 | 4.99e−07 | 1.3651 | 3.4832 | 0.9998 | 2.35e−07 |
DoF |
h
|
\(\Vert \varvec{u}-\varvec{u}_h\Vert _{\mathrm {H}}\)
|
\(\Vert \varvec{\omega }-\varvec{\omega }_h\Vert _{\mathrm {Z}}\)
|
\(\Vert p-p_h\Vert _{0,\varOmega }\)
| rate
|
\(\Vert \mathop {\mathrm {div}}\nolimits \varvec{u}_h\Vert _{\ell ^\infty }\)
|
---|---|---|---|---|---|---|
\(k=0\)
| ||||||
34 | 0.7071 | 5.35e−16 | 9.88e−17 | 0.1918 | – | 1.03e−15 |
114 | 0.3536 | 6.15e−17 | 2.56e−17 | 0.1073 | 0.8384 | 1.78e−16 |
418 | 0.1768 | 7.61e−16 | 1.64e−16 | 0.0551 | 0.9597 | 2.53e−15 |
1602 | 0.0884 | 9.08e−15 | 3.11e−15 | 0.0277 | 0.9902 | 4.13e−14 |
6274 | 0.0442 | 1.90e−13 | 4.16e−14 | 0.0139 | 0.9975 | 2.59e−12 |
24834 | 0.0221 | 7.34e−12 | 6.41e−13 | 0.0069 | 0.9994 | 1.61e−10 |
98818 | 0.0111 | 1.98e−10 | 5.47e−12 | 0.0038 | 0.9998 | 9.79e−09 |
\(k=1\)
| ||||||
98 | 0.7071 | 2.34e−16 | 4.41e−16 | 0.0450 | – | 2.42e−16 |
354 | 0.3536 | 7.46e−15 | 1.36e−14 | 0.0120 | 1.8970 | 2.57e−14 |
1346 | 0.1768 | 7.04e−15 | 1.65e−14 | 0.0030 | 1.9752 | 3.73e−14 |
5250 | 0.0884 | 7.31e−14 | 4.94e−14 | 0.0007 | 1.9944 | 7.71e−13 |
20738 | 0.0442 | 8.57e−12 | 1.40e−13 | 0.0002 | 1.9982 | 2.62e−09 |
82434 | 0.0221 | 3.94e−12 | 5.02e−11 | 4.8e−05 | 2.0001 | 9.57e−08 |
328706 | 0.0111 | 1.98e−11 | 2.56e−11 | 1.2e−05 | 2.0000 | 5.67e−08 |
DoF |
h
|
\(|(\varvec{u}-\varvec{u}_h,\varvec{\omega }-\varvec{\omega }_h,p-p_h)|_{{\mathcal {A}}}\)
| rate
|
\(\Vert p-p_h\Vert _{0,\varOmega }\)
| rate
|
---|---|---|---|---|---|
\(k=0\)
| |||||
65 | 0.7071 | 0.8031 | – | 0.4623 | – |
257 | 0.3536 | 0.4321 | 0.8893 | 0.2298 | 0.8384 |
1025 | 0.1768 | 0.2343 | 0.8827 | 0.1276 | 0.8438 |
4097 | 0.0884 | 0.1217 | 0.9448 | 0.0652 | 0.9668 |
16385 | 0.0442 | 0.0616 | 0.9822 | 0.0326 | 1.0010 |
65537 | 0.0221 | 0.0302 | 1.0231 | 0.0163 | 1.0005 |
\(k=1\)
| |||||
145 | 0.7071 | 0.4787 | – | 0.1847 | – |
577 | 0.3536 | 0.1496 | 1.6760 | 0.0529 | 1.8033 |
2305 | 0.1768 | 0.0366 | 2.0297 | 0.0133 | 1.9914 |
9217 | 0.0884 | 0.0089 | 2.0413 | 0.0033 | 2.0220 |
36865 | 0.0442 | 0.0021 | 2.0320 | 0.0008 | 2.0361 |
134696 | 0.0221 | 0.0005 | 2.0034 | 0.0002 | 2.0049 |
\(k=2\)
| |||||
257 | 0.7071 | 0.1882 | – | 0.0534 | – |
1025 | 0.3536 | 0.0319 | 2.5864 | 0.0111 | 2.2722 |
4097 | 0.1768 | 0.0042 | 2.9180 | 0.0013 | 3.0093 |
16385 | 0.0884 | 0.0005 | 3.0362 | 0.0002 | 3.1105 |
65537 | 0.0442 | 6.16e−5 | 3.0640 | 1.81e−5 | 3.1510 |
268049 | 0.0221 | 1.03e−5 | 2.9973 | 2.78e−6 | 3.0076 |
DoF |
h
|
\(|(\varvec{u}-\varvec{u}_h,\varvec{\omega }-\varvec{\omega }_h,p-p_h)|_{{\mathcal {A}}}\)
| rate
|
\(\Vert p-p_h\Vert _{0,\varOmega }\)
| rate
|
---|---|---|---|---|---|
\(\nu =0.01\)
| |||||
65 | 0.7071 | 0.5059 | – | 0.2034 | – |
257 | 0.3536 | 0.2884 | 0.8105 | 0.1264 | 0.6864 |
1025 | 0.1768 | 0.1642 | 0.8126 | 0.0842 | 0.7585 |
4097 | 0.0884 | 0.0902 | 0.8632 | 0.0507 | 0.7929 |
16385 | 0.0442 | 0.0473 | 0.9315 | 0.0274 | 0.8878 |
65537 | 0.0221 | 0.0244 | 0.9514 | 0.0140 | 0.9606 |
\(\nu =0.0001\)
| |||||
65 | 0.7071 | 0.3787 | – | 0.1963 | – |
257 | 0.3536 | 0.1882 | 1.0091 | 0.1102 | 0.8322 |
1025 | 0.1768 | 0.0796 | 1.2426 | 0.0562 | 0.9709 |
4097 | 0.0884 | 0.0363 | 1.1314 | 0.0285 | 0.9916 |
16385 | 0.0442 | 0.0176 | 1.0423 | 0.0141 | 0.9974 |
65537 | 0.0221 | 0.0087 | 1.0154 | 0.0071 | 0.9992 |
\(\nu =0.000001\)
| |||||
65 | 0.7071 | 0.3434 | – | 0.1967 | – |
257 | 0.3536 | 0.1490 | 1.2025 | 0.1088 | 0.8488 |
1025 | 0.1768 | 0.0586 | 1.3461 | 0.0553 | 0.9764 |
4097 | 0.0884 | 0.0239 | 1.2931 | 0.0278 | 0.9933 |
16385 | 0.0442 | 0.0108 | 1.1437 | 0.0135 | 0.9982 |
65537 | 0.0221 | 0.0051 | 1.0654 | 0.0069 | 0.9996 |