Preliminary
computations
Daniele Boffi
1D Laplace
2D Laplace
1D mixed
Laplace
2D mixed
Laplace
Preliminary computations
Daniele Boffi
Dipartimento di Matematica “F. Casorati”, Università di Pavia
http://www-dimat.unipv.it/boffi
Preliminary
computations
1D Laplace: P1
Daniele Boffi
1D Laplace
P1
P2
Spectral method
2D Laplace
1D mixed
Laplace
2D mixed
Laplace
1
4
9
16
25
n=8
n = 16
n = 32
1.0129160450588
4.2095474481529
10.0802909335883
19.4536672593288
33.2628304890884
1.0032168743567
4.0516641802355
9.2631305555446
16.8381897926118
27.0649225609802
1.0008034482562
4.0128674974272
9.0652448637285
16.2066567209423
25.5059230069702
Preliminary
computations
1D Laplace: P1
Daniele Boffi
1D Laplace
P1
P2
Spectral method
2D Laplace
1D mixed
Laplace
2D mixed
Laplace
1
4
9
16
25
1
4
9
16
25
n=8
n = 16
n = 32
1.0129160450588
4.2095474481529
10.0802909335883
19.4536672593288
33.2628304890884
1.0032168743567
4.0516641802355
9.2631305555446
16.8381897926118
27.0649225609802
1.0008034482562
4.0128674974272
9.0652448637285
16.2066567209423
25.5059230069702
n = 64
n = 128
n = 256
1.0002008137390
4.0032137930241
9.0162763381719
16.0514699897078
25.1257489536113
1.0000502004122
4.0008032549556
9.0040668861371
16.0128551720960
25.0313903532369
1.0000125499161
4.0002008016414
9.0010165838380
16.0032130198251
25.0078446408520
Preliminary
computations
P2
Daniele Boffi
1D Laplace
P1
P2
Spectral method
2D Laplace
1D mixed
Laplace
2D mixed
Laplace
Computed (rate)
1
4
9
16
25
36
49
64
81
100
#
n=8
n = 16
1.0000
1.0000
4.0020
4.0001
9.0225
9.0015
16.1204 16.0082
25.4327 25.0307
37.1989 36.0899
51.6607 49.2217
64.8456 64.4814
95.7798 81.9488
124.9301 101.7308
15
31
n = 32
(4.0)
1.0000
(4.0)
4.0000
(3.9)
9.0001
(3.9) 16.0005
(3.8) 25.0020
(3.7) 36.0059
(3.6) 49.0148
(0.8) 64.0328
(4.0) 81.0659
(3.8) 100.1229
63
n = 64
(4.0)
1.0000
(4.0)
4.0000
(4.0)
9.0000
(4.0) 16.0000
(3.9) 25.0001
(3.9) 36.0004
(3.9) 49.0009
(3.9) 64.0021
(3.8) 81.0042
(3.8) 100.0080
127
n = 128
(4.0)
1.0000
(4.0)
4.0000
(4.0)
9.0000
(4.0) 16.0000
(4.0) 25.0000
(4.0) 36.0000
(4.0) 49.0001
(4.0) 64.0001
(4.0) 81.0003
(3.9) 100.0005
255
(4.0)
(4.0)
(4.0)
(4.0)
(4.0)
(4.0)
(4.0)
(4.0)
(4.0)
(4.0)
Preliminary
computations
Spectral method
Daniele Boffi
1D Laplace
P1
P2
Spectral method
2D Laplace
1D mixed
Laplace
2D mixed
Laplace
Convergence of fifth eigenvalue
p
DOF
Computed
7
8
9
10
11
12
13
14
15
16
17
18
19
20
5
6
7
8
9
10
11
12
13
14
15
16
17
18
35.5593555378041
35.5593555378041
25.7779168651921
25.7779168651921
25.0306605127133
25.0306605127132
25.0004945052929
25.0004945052929
25.0000037734250
25.0000037734250
25.0000000156754
25.0000000156756
25.0000000000389
25.0000000000389
Preliminary
computations
Spectral method (cont’ed)
Daniele Boffi
1D Laplace
P1
P2
Spectral method
Exponential convergence
2D Laplace
0
1D mixed
Laplace
10
2D mixed
Laplace
−5
10
−10
10
−15
10
5
10
15
20
Preliminary
computations
2D Laplace: P1
Daniele Boffi
1D Laplace
2D Laplace
P1
L-shaped domain
non-conf. P1
Unstructured mesh
1D mixed
Laplace
2D mixed
Laplace
Computed (rate)
2
5
5
8
10
10
13
13
17
17
#
N=4
2.2468
6.5866
6.6230
10.2738
12.7165
14.3630
19.7789
24.2262
34.0569
9
N=8
2.0463 (2.4)
5.2732 (2.5)
5.2859 (2.5)
8.7064 (1.7)
11.0903 (1.3)
11.1308 (1.9)
14.8941 (1.8)
14.9689 (2.5)
20.1284 (2.4)
20.2113
56
N
2.0106
5.0638
5.0643
8.1686
10.2550
10.2595
13.4370
13.4435
17.7468
17.7528
257
= 16
(2.1)
(2.1)
(2.2)
(2.1)
(2.1)
(2.1)
(2.1)
(2.2)
(2.1)
(2.1)
N
2.0025
5.0154
5.0156
8.0402
10.0610
10.0622
13.1046
13.1053
17.1771
17.1798
1106
= 32
(2.1)
(2.0)
(2.0)
(2.1)
(2.1)
(2.1)
(2.1)
(2.1)
(2.1)
(2.1)
N
2.0006
5.0038
5.0038
8.0099
10.0152
10.0153
13.0258
13.0258
17.0440
17.0443
4573
= 64
(2.0)
(2.0)
(2.0)
(2.0)
(2.0)
(2.0)
(2.0)
(2.0)
(2.0)
(2.0)
Preliminary
computations
Multiple eigenfunctions
Daniele Boffi
1D Laplace
2D Laplace
P1
L-shaped domain
non-conf. P1
1D mixed
Laplace
2D mixed
Laplace
3
3
3
2.5
2.5
2.5
2
2
2
1.5
1.5
1.5
1
1
1
0.5
0.5
0.5
0
0
0
0.5
1
1.5
2
2.5
3
0
0
0.5
1
1.5
2
2.5
3
3
3
3
2.5
2.5
2.5
2
2
2
1.5
1.5
1.5
1
1
1
0.5
0.5
0.5
0
0
0
0.5
1
1.5
2
2.5
3
0
0.5
1
1.5
2
2.5
3
0
0.5
1
1.5
2
2.5
3
0
0
0.5
1
1.5
2
2.5
3
Preliminary
computations
Daniele Boffi
Multiple eigenfunctions (cont’ed)
1D Laplace
2D Laplace
P1
L-shaped domain
non-conf. P1
1D mixed
Laplace
2D mixed
Laplace
Exact solutions
Preliminary
computations
Daniele Boffi
1D Laplace
2D Laplace
P1
L-shaped domain
non-conf. P1
1D mixed
Laplace
2D mixed
Laplace
Multiple eigenfunctions (cont’ed)
Uniform mesh
3
3
2.5
2.5
2
2
1.5
1.5
1
1
0.5
0.5
0
0
0
0.5
1
1.5
2
2.5
3
0
0.5
1
1.5
2
2.5
3
0
0.5
1
1.5
2
2.5
3
Uniform mesh (reversed)
3
3
2.5
2.5
2
2
1.5
1.5
1
1
0.5
0.5
0
0
0
0.5
1
1.5
2
2.5
3
Preliminary
computations
Uniform mesh
Daniele Boffi
1D Laplace
2D Laplace
P1
L-shaped domain
non-conf. P1
Computed (rate)
1D mixed
Laplace
2D mixed
Laplace
2
5
5
8
10
10
13
13
17
17
#
N=4
2.3168
6.3387
7.2502
12.2145
15.5629
16.7643
20.8965
26.0989
32.4184
9
N=8
2.0776 (2.0)
5.3325 (2.0)
5.5325 (2.1)
9.1826 (1.8)
11.5492 (1.8)
11.6879 (2.0)
15.2270 (1.8)
17.0125 (1.7)
21.3374 (1.8)
21.5751
49
N
2.0193
5.0829
5.1302
8.3054
10.3814
10.3900
13.5716
13.9825
18.0416
18.0705
225
= 16
(2.0)
(2.0)
(2.0)
(2.0)
(2.0)
(2.1)
(2.0)
(2.0)
(2.1)
(2.1)
N
2.0048
5.0207
5.0324
8.0769
10.0949
10.0955
13.1443
13.2432
17.2562
17.2626
961
= 32
(2.0)
(2.0)
(2.0)
(2.0)
(2.0)
(2.0)
(2.0)
(2.0)
(2.0)
(2.0)
N
2.0012
5.0052
5.0081
8.0193
10.0237
10.0237
13.0362
13.0606
17.0638
17.0653
3969
= 64
(2.0)
(2.0)
(2.0)
(2.0)
(2.0)
(2.0)
(2.0)
(2.0)
(2.0)
(2.0)
Preliminary
computations
Crisscross mesh
Daniele Boffi
1D Laplace
2D Laplace
Computed (rate)
P1
L-shaped domain
non-conf. P1
1D mixed
Laplace
2D mixed
Laplace
2
5
5
8
10
10
13
13
17
17
18
20
20
25
25
#
N=4
2.0880
5.6811
5.6811
9.4962
12.9691
12.9691
17.1879
17.1879
25.1471
38.9073
38.9073
38.9073
38.9073
38.9073
38.9073
25
N=8
2.0216 (2.0)
5.1651 (2.0)
5.1651 (2.0)
8.3521 (2.1)
10.7578 (2.0)
10.7578 (2.0)
14.0237 (2.0)
14.0237 (2.0)
19.3348 (1.8)
19.3348 (3.2)
19.8363 (3.5)
22.7243 (2.8)
22.7243 (2.8)
28.7526 (1.9)
28.7526 (1.9)
113
N
2.0054
5.0408
5.0408
8.0863
10.1865
10.1865
13.2489
13.2489
17.5733
17.5733
18.4405
20.6603
20.6603
25.8940
25.8940
481
= 16
(2.0)
(2.0)
(2.0)
(2.0)
(2.0)
(2.0)
(2.0)
(2.0)
(2.0)
(2.0)
(2.1)
(2.0)
(2.0)
(2.1)
(2.1)
N
2.0013
5.0102
5.0102
8.0215
10.0464
10.0464
13.0617
13.0617
17.1423
17.1423
18.1089
20.1634
20.1634
25.2201
25.2201
1985
= 32
(2.0)
(2.0)
(2.0)
(2.0)
(2.0)
(2.0)
(2.0)
(2.0)
(2.0)
(2.0)
(2.0)
(2.0)
(2.0)
(2.0)
(2.0)
N
2.0003
5.0025
5.0025
8.0054
10.0116
10.0116
13.0154
13.0154
17.0355
17.0355
18.0271
20.0407
20.0407
25.0548
25.0548
8065
= 64
(2.0)
(2.0)
(2.0)
(2.0)
(2.0)
(2.0)
(2.0)
(2.0)
(2.0)
(2.0)
(2.0)
(2.0)
(2.0)
(2.0)
(2.0)
Preliminary
computations
L-shaped domain
Daniele Boffi
1D Laplace
2D Laplace
P1
L-shaped domain
non-conf. P1
P1 elements (Neumann boundary conditions)
1D mixed
Laplace
Computed (rate)
2D mixed
Laplace
0
1.48
3.53
9.87
9.87
11.39
#
N=4
N=8
-0.0000
0.0000
1.6786
1.5311 (1.9)
3.8050
3.5904 (2.3)
12.2108 10.2773 (2.5)
12.5089 10.3264 (2.5)
13.9526 12.0175 (2.0)
20
65
N
-0.0000
1.4946
3.5472
9.9692
9.9823
11.5303
245
= 16
(1.5)
(2.1)
(2.0)
(2.0)
(2.2)
N
-0.0000
1.4827
3.5373
9.8935
9.8979
11.4233
922
= 32
N
-0.0000
(1.4)
1.4783
(2.0)
3.5348
(2.1)
9.8755
(2.0)
9.8767
(2.1) 11.3976
3626
= 64
(1.4)
(2.0)
(2.0)
(2.0)
(2.1)
Preliminary
computations
Nonconforming P1
Daniele Boffi
1D Laplace
2D Laplace
P1
L-shaped domain
non-conf. P1
Computed (rate)
1D mixed
Laplace
2D mixed
Laplace
2
5
5
8
10
10
13
13
17
17
#
N=4
1.9674
4.4508
4.7270
7.2367
8.5792
9.0237
9.8284
9.9107
10.4013
11.2153
40
N=8
1.9850 (1.1)
4.9127 (2.7)
4.9159 (1.7)
7.7958 (1.9)
9.6553 (2.0)
9.6663 (1.5)
12.4011 (2.4)
12.4637 (2.5)
15.9559 (2.7)
16.0012 (2.5)
197
N
1.9966
4.9787
4.9790
7.9434
9.9125
9.9197
12.8534
12.8561
16.7485
16.7618
832
= 16
(2.1)
(2.0)
(2.0)
(1.9)
(2.0)
(2.1)
(2.0)
(1.9)
(2.1)
(2.1)
N
1.9992
4.9949
4.9949
7.9870
9.9792
9.9796
12.9654
12.9655
16.9407
16.9409
3443
= 32
(2.0)
(2.1)
(2.0)
(2.1)
(2.1)
(2.0)
(2.1)
(2.1)
(2.1)
(2.0)
N
1.9998
4.9987
4.9987
7.9967
9.9949
9.9950
12.9914
12.9914
16.9853
16.9854
13972
= 64
(2.0)
(2.0)
(2.0)
(2.0)
(2.0)
(2.0)
(2.0)
(2.0)
(2.0)
(2.0)
Preliminary
computations
Mixed Laplace
Daniele Boffi
1D Laplace
2D Laplace
1D mixed
Laplace
1D mixed Laplace: P1-P1 scheme
P1-P1
P1-P0
P2-P0
Computed (rate)
2D mixed
Laplace
1
4
9
16
25
36
49
N=8
0.0000
1.0001
3.9660
7.4257
8.7603
14.8408
16.7900
38.7154
39.0906
N
-0.0000
1.0000
3.9981
8.5541
8.9873
15.9501
24.5524
29.7390
35.0393
46.7793
= 16
(4.1)
(4.2)
(4.2)
(4.5)
(4.2)
(1.7)
N
-0.0000
1.0000
3.9999
8.8854
8.9992
15.9971
24.9780
34.2165
35.9492
48.8925
= 32
(4.0)
(4.0)
(4.1)
(4.1)
(4.3)
(4.2)
(4.4)
N
-0.0000
1.0000
4.0000
8.9711
9.0000
15.9998
24.9987
35.5415
35.9970
48.9937
= 64
(4.0)
(4.0)
(4.0)
(4.0)
(4.1)
(4.1)
(4.1)
N = 128
-0.0000
1.0000 (4.0)
4.0000 (4.0)
8.9928
9.0000 (4.0)
16.0000 (4.0)
24.9999 (4.0)
35.8846
35.9998 (4.0)
48.9996 (4.0)
Preliminary
computations
P1-P1 (cont’ed)
Daniele Boffi
1D Laplace
2D Laplace
1D mixed
Laplace
P1-P1
P1-P0
P2-P0
2D mixed
Laplace
Results for N = 1000
Exact
1
4
9
16
25
36
49
Computed
N = 1000
-0.0000000000
1.0000000000
3.9999999999
8.9998815658
8.9999999992
15.9999999971
24.9999999784
35.9981051039
35.9999999495
48.9999998977
Preliminary
computations
P1-P0 scheme
Daniele Boffi
1D Laplace
2D Laplace
1D mixed
Laplace
Computed (rate)
P1-P1
P1-P0
P2-P0
2D mixed
Laplace
1
4
9
16
25
36
49
64
81
100
#
n=8
1.0129
4.2095
10.0803
19.4537
33.2628
51.3724
69.5582
77.8147
n = 16
1.0032
4.0517
9.2631
16.8382
27.0649
40.3212
57.0672
77.8147
103.0473
133.0513
8
16
(2.0)
(2.0)
(2.0)
(2.0)
(2.0)
(1.8)
(1.3)
(0.0)
n = 32
1.0008
4.0129
9.0652
16.2067
25.5059
37.0525
50.9572
67.3528
86.3943
108.2597
32
n = 64
(2.0)
1.0002
(2.0)
4.0032
(2.0)
9.0163
(2.0)
16.0515
(2.0)
25.1257
(2.0)
36.2610
(2.0)
49.4840
(2.0)
64.8266
(2.0)
82.3258
(2.0) 102.0237
64
(2.0)
(2.0)
(2.0)
(2.0)
(2.0)
(2.0)
(2.0)
(2.0)
(2.0)
(2.0)
n = 128
1.0001
4.0008
9.0041
16.0129
25.0314
36.0651
49.1206
64.2059
81.3299
100.5030
128
(2.0)
(2.0)
(2.0)
(2.0)
(2.0)
(2.0)
(2.0)
(2.0)
(2.0)
(2.0)
Preliminary
computations
P2-P0 scheme
Daniele Boffi
1D Laplace
2D Laplace
1D mixed
Laplace
Computed (rate with respect to 6λ)
P1-P1
P1-P0
P2-P0
2D mixed
Laplace
1
4
9
16
25
36
49
64
81
100
#
n=8
5.7061
19.8800
36.7065
51.8764
63.6140
71.6666
76.3051
77.8147
n = 16
5.9238
22.8245
48.3798
79.5201
113.1819
146.8261
178.6404
207.5058
232.8461
254.4561
8
16
(1.9)
(1.8)
(1.6)
(1.4)
(1.2)
(1.1)
(0.9)
(0.8)
n = 32
5.9808
23.6953
52.4809
91.2978
138.8165
193.5192
253.8044
318.0804
384.8425
452.7277
32
n = 64
(2.0)
5.9952 (2.0)
(1.9)
23.9231 (2.0)
(1.9)
53.6123 (2.0)
(1.8)
94.7814 (1.9)
(1.7) 147.0451 (1.9)
(1.6) 209.9235 (1.9)
(1.5) 282.8515 (1.9)
(1.4) 365.1912 (1.8)
(1.3) 456.2445 (1.8)
(1.2) 555.2659 (1.7)
64
n = 128
5.9988
23.9807
53.9026
95.6925
149.2506
214.4494
291.1344
379.1255
478.2172
588.1806
128
(2.0)
(2.0)
(2.0)
(2.0)
(2.0)
(2.0)
(2.0)
(1.9)
(1.9)
(1.9)
Preliminary
computations
Raviart–Thomas element
Daniele Boffi
1D Laplace
2D Laplace
Unstructured mesh
1D mixed
Laplace
2D mixed
Laplace
Computed (rate)
Raviart–Thomas
2
5
5
8
10
10
13
13
17
17
#
N=4
2.0138
4.8696
4.8868
8.6905
9.7590
11.4906
11.9051
12.7210
13.5604
14.1813
32
N=8
1.9989 (3.6)
4.9920 (4.0)
4.9952 (4.5)
7.9962 (7.5)
9.9725 (3.1)
9.9911 (7.4)
12.9250 (3.9)
12.9631 (2.9)
16.8450 (4.5)
16.9659 (6.4)
142
N
1.9997
5.0000
5.0006
7.9974
9.9980
10.0007
12.9917
12.9950
16.9848
16.9946
576
= 16
(1.7)
(8.0)
(3.0)
(0.6)
(3.8)
(3.7)
(3.2)
(2.9)
(3.4)
(2.7)
N = 32
1.9999 (2.7)
4.9999 (-2.1)
5.0000 (5.8)
7.9995 (2.5)
9.9992 (1.3)
10.0005 (0.4)
12.9998 (5.4)
13.0000 (7.5)
16.9992 (4.3)
17.0009 (2.6)
2338
N
2.0000
5.0000
5.0000
7.9999
9.9999
10.0001
12.9999
13.0000
16.9999
17.0000
9400
= 64
(2.8)
(3.7)
(2.6)
(2.2)
(3.2)
(2.4)
(1.8)
(1.1)
(2.5)
(5.5)
Preliminary
computations
Raviart–Thomas element (cont’ed)
Daniele Boffi
1D Laplace
2D Laplace
Uniform mesh
1D mixed
Laplace
2D mixed
Laplace
Computed (rate)
Raviart–Thomas
2
5
5
8
10
10
13
13
17
17
#
N=4
2.1048
5.9158
5.9158
9.7268
13.8955
13.8955
17.7065
17.7065
20.5061
20.5061
16
N=8
2.0258 (2.0)
5.2225 (2.0)
5.2225 (2.0)
8.4191 (2.0)
11.0932 (1.8)
11.0932 (1.8)
14.2898 (1.9)
14.2898 (1.9)
20.1606 (0.1)
20.4666 (0.0)
64
N
2.0064
5.0549
5.0549
8.1033
10.2663
10.2663
13.3148
13.3148
17.8414
17.8414
256
= 16
(2.0)
(2.0)
(2.0)
(2.0)
(2.0)
(2.0)
(2.0)
(2.0)
(1.9)
(2.0)
N
2.0016
5.0137
5.0137
8.0257
10.0660
10.0660
13.0781
13.0781
17.2075
17.2075
1024
= 32
(2.0)
(2.0)
(2.0)
(2.0)
(2.0)
(2.0)
(2.0)
(2.0)
(2.0)
(2.0)
N
2.0004
5.0034
5.0034
8.0064
10.0165
10.0165
13.0195
13.0195
17.0517
17.0517
4096
= 64
(2.0)
(2.0)
(2.0)
(2.0)
(2.0)
(2.0)
(2.0)
(2.0)
(2.0)
(2.0)