Monge-Ampére

4.3 Monge-Ampére

Regularization Monge-Ampére Equation:

4²u + det D²u = f; in u = gon @

4u = on @

where f and g are obtained from a given analytical solution u. We use BCIZ element to approximate the Monge-Ampére equation. We compute the Poisson equation on different parameter with xed mesh size h. The boudary condition are Dirchlet type. In this section, we provide several 2-D numerical experiments of BCIZ element. And the initial condition is given by zero. The start from 1 to h²:

4.3 Monge-Ampére 54

4.3.1 Example:

This test, we calculus ku⁰ u_hk for xed mesh size h = 2 ⁸;while varying in order to approximate ku⁰ u k : We use BCIZ element and set to solve problem (3:71) with the analytical solution u = x⁴+ y² ; f = 24x²and g = x⁴+ y²;Our calculation domain is [0; 1] [0; 1] :

solution error

ku⁰ u_hk₁ ku⁰ u_hkL² ku⁰ u_hkH² iter

1 3.05E-1 1.61E-1 5.32 6

2 ² 2.30E-1 1.21E-1 4.67 10

2 ⁴ 1.13E-1 5.71E-2 3.63 10

2 ⁶ 4.23E-2 1.90E-2 2.71 8

2 ⁸ 1.45E-2 5.67E-3 1.99 8

2 ¹⁰ 4.50E-3 1.60E-3 1.44 8

2 ¹² 1.29E-3 4.33E-4 1.03 9

2 ¹⁴ 3.48E-4 1.13E-4 7.33E-1 10

Table 5. Change of ku⁰ u_hk w.r.t. (h = 2 ⁸)

4.3 Monge-Ampére 55

k^{u0 uh}kL2 k^{u0 uh}kH2

p4

1 0.160842924 5.319207667 2 ² 0.482036757 6.609013597 2 ⁴ 0.913779843 7.268095576 2 ⁶ 1.218354047 7.670044224 2 ⁸ 1.450567862 7.955838131 2 ¹⁰ 1.637202829 8.133261803 2 ¹² 1.772707203 8.235401948 2 ¹⁴ 1.85193153 8.290262971 Table 6. Change of ku⁰ u_hk w.r.t. (h = 2 ⁸)

4.3 Monge-Ampére 56

4.3.2 Example:

This test, we calculus ku⁰ u_hk for xed mesh size h = 2 ⁸;while varying in order to approximate ku⁰ u k : We use BCIZ element and set to solve problem (3:71) with the analytical solution u = 20x⁶ + y⁶ ; f = 18000x⁴y⁴ and g = 20x⁶+ y⁶;Our calculation domain is [0; 1] [0; 1] :

solution error

ku⁰ u_hk₁ ku⁰ u_hkL² ku⁰ u_hkH² iter

4 6.43 2.88 177.28 5

1 5.22 2.17 167.40 9

2 ² 3.31 1.19 148.99 10

2 ⁴ 1.79 6.20E-1 125.31 10

2 ⁶ 8.72E-1 2.57E-1 101.97 10

2 ⁸ 4.02E-1 9.17E-2 82.10 10

2 ¹⁰ 1.80E-1 3.22E-2 65.25 11

2 ¹² 7.93E-2 1.11E-2 51.21 13

2 ¹⁴ 3.45E-2 3.74E-3 39.77 20

Table 7. Change of ku uhk w.r.t. (h = 2 ⁸)

4.3 Monge-Ampére 57

k^{u0 uh}kL2 k^{u0 uh}kH2

p4

4 0.720510991 125.3567907 1 2.165121991 167.3973101 2 ² 4.758836554 210.7002103 2 ⁴ 9.926086663 250.6286892 2 ⁶ 16.4590354 288.4238466 2 ⁸ 23.47587569 328.3874835 2 ¹⁰ 32.93384144 369.1038869 2 ¹² 45.33800541 409.6430813 2 ¹⁴ 61.24535142 449.9969415 Table 8. Change of ku uhk w.r.t. (h = 2 ⁸)

4.3 Monge-Ampére 58

4.3.3 Example:

This test, we calculus ku⁰ u_hk for xed mesh size h = 2 ⁸;while varying in order to approximate ku⁰ u k : We use BCIZ element and set to solve problem (3:71) with the analytical solution u = e^x2+y22 ; f = (1 + x²+ y²) e^x2+y2 and g = e^x2+y22 ;Our calculation domain is [0; 1] [0; 1] :

solution error

ku⁰ u_hk₁ ku⁰ u_hkL² ku⁰ u_hkH² iter

1 1.78E-1 1.01E-1 3.03 29

2 ² 1.41E-1 8.17E-2 2.72 48

2 ⁴ 7.14E-2 4.45E-2 2.13 38

2 ⁶ 2.26E-2 1.56E-2 1.57 9

2 ⁸ 6.30E-3 4.52E-3 1.13 8

2 ¹⁰ 1.81E-3 1.22E-3 8.00E-1 8

2 ¹² 5.00E-4 3.16E-4 5.67E-1 8

2 ¹⁴ 1.33E-4 8.00E-5 4.01E-1 9

Table 9. Change of ku uhk w.r.t. (h = 2 ⁸)

4.3 Monge-Ampére 59

k^{u0 uh}kL2 k^{u0 uh}kH2

p4

1 0.100518486 3.026105949 2 ² 0.32688765 3.840967403 2 ⁴ 0.71238757 4.259311736 2 ⁶ 0.99970787 4.434163657 2 ⁸ 1.157804247 4.506118399 2 ¹⁰ 1.24700059 4.527808719 2 ¹² 1.294427725 4.534697517 2 ¹⁴ 1.309904805 4.53757113

Table 10. Change of ku uhk w.r.t. (h = 2 ⁸)

4.3 Monge-Ampére 60

4.3.4 Example:

This test, we calculus ku⁰ u_hk for xed mesh size h = 2 ⁸;while varying in order to approximate ku⁰ u k : We use BCIZ element and set to solve problem (3:71) with the analytical solution u = ^2p2(^x2+y2)³⁴

3 ; f = p ¹

x²+y² and g = ^2p2(^x2+y2)³⁴

3 ;Our calculation domain is [0; 1] [0; 1] :Where f has a singular point at (0; 0) :

solution error

ku⁰ u_hk₁ ku⁰ u_hkL² ku⁰ u_hkH² iter

1 1.41E-1 7.89E-2 2.16 5

2 ² 1.23E-1 6.93E-2 2.01 19

2 ⁴ 7.59E-2 4.48E-2 1.64 9

2 ⁶ 2.78E-2 1.81E-2 1.22 10

2 ⁸ 8.01E-3 5.57E-3 8.95E-1 8

2 ¹⁰ 2.12E-3 1.55E-3 6.51E-1 8

2 ¹² 5.57E-4 4.08E-4 4.68E-1 9

2 ¹⁴ 1.44E-4 1.04E-4 3.34E-1 11

Table 11. Change of ku uhk w.r.t. (h = 2 ⁸)

4.3 Monge-Ampére 61

k^{u0 uh}kL2 k^{u0 uh}kH2

p4

1 0.078906702 2.164987734 2 ² 0.277330363 2.843363264 2 ⁴ 0.717285822 3.275360039 2 ⁶ 1.15590166 3.442870121 2 ⁸ 1.425652095 3.57845558 2 ¹⁰ 1.583580447 3.680240848 2 ¹² 1.671089711 3.747674023 2 ¹⁴ 1.709884392 3.773972909 Table 12. Change of ku uhk w.r.t. (h = 2 ⁸)

4.3 Monge-Ampére 62

4.3.5 Example:

This test, we calculus ku⁰ u_hk for xed mesh size h = 2 ⁸;while varying in order to approximate ku⁰ u k : We use BCIZ element and set to solve problem (3:71) with the analytical solution u =p

x²+ y²; f = 0 if (x; y) 6= (0; 0)

? if (x; y) = (0; 0) and g =p

x²+ y²;Our calculation domain is [ 1; 1] [ 1; 1] :Where f has a singular point at (0; 0), and our guess the value of f at (0; 0) is 3 :

solution error

ku⁰ u_hk₁ ku⁰ u_hkL² ku⁰ u_hkH² iter

1 8.17E-1 6.00E-1 8.65 5

2 ² 5.51E-1 3.72E-1 8.82 9

2 ⁴ 2.48E-1 1.42E-1 9.27 10

2 ⁶ 9.77E-2 5.31E-2 9.72 11

2 ⁸ 4.151E-2 2.56E-2 10.09 14

2 ¹⁰ 2.08E-2 1.49E-2 10.32 19

2 ¹² 9.28E-3 6.92E-3 10.51 28

2 ¹³ 3.59E-3 1.95E-3 10.66 21

Table 13. Change of ku uhk w.r.t. (h = 1=127)

4.3 Monge-Ampére 63

k^{u0 uh}kL2 k^{u0 uh}kH2

p4

1 0.599947604 8.652542689 2 ² 1.48793695 12.4786026 2 ⁴ 2.278636795 18.53711229 2 ⁶ 3.398612852 27.50569621 2 ⁸ 6.546636298 40.36922514 2 ¹⁰ 15.30464406 58.38278827 2 ¹² 28.35360433 84.08190819 2 ¹³ 16.00805235 101.4092367

Table 14. Change of ku uhk w.r.t. (h = 1=127)

4.3 Monge-Ampére 64

Compared with Feng's and Oberman's result:

case1: u = e^x2+y22 ; f = (1 + x² + y²) e^x2+y2

Ours Feng's

ku u_hkL² ku u_hkL²

2 ¹ 0.093580805 0:5 0.038717 2 ² 0.081721912 0:25 0.040988 2 ³ 0.064370429 0:1 0.032218 2 ⁴ 0.044524223 0:05 0.022259 2 ⁶ 0.015620435 0:0125 0.007817 2 ⁹ 0.002361013 0:0025 0.001864 2 ¹¹ 0.00062253 0:0005 0.000405

Ours Oberman's

N iter N M1 M2

128 69 121 31965 1205

4.3 Monge-Ampére 65

case 2: u = p

x²+ y²; f = 0 if (x; y) 6= (0; 0)

? if (x; y) = (0; 0)

Ours Feng's

ku u_hkL² ku u_hkL²

2 ¹ 0.504058 2 ² 0.371984 2 ³ 0.239381

2 ⁴ 0.142415 none

2 ⁶ 0.053103 2 ⁹ 0.014946 2 ¹¹ 0.001954

Ours Oberman's

N iter N M1 M2

128 117 121 36396 10486

Under same accuracy, iteration number is less than other group. And the case have singularty can be achieved with high ef cient computing.

Chapter 5 Conclusion

1. The error of ku uhkL² of the Monge-Ampére is O ( ) from test cases. The error of ku u_hkH² of the Monge-Ampére is O (p⁴

)from test cases.

2. In numerical simulation of the elliptic regularization Monge-Ampére, the h²; where h is mesh size.

3. In the singular case, we can shift the grid point such that the singular point locate in a element.

66

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在文檔中 Monge-Amp□re方程的數值方法與其在非成像光學上的應用 (頁 59-0)