Derive the element matrix

2.6 Derive the element matrix

The variation formulation of Possin equation Z

on each element Ki the global coordinate x; y transfor into local ₁; ₂; ₃ the linear La-grange interpolation of w in the local interpolant w = N1'₁+N₂'₂+N₃'₃ =

the local element matrix isR

eir ^{; T}Z^TZr ^; jJj d d

Chapter 3

Numerical Method of Monge-Ampére Equation

We follow the Feng's method, that is adding a vanishing biharmonic term such that the fully non-linear Monge-Ampére equation become regular. The elliptic regularization Monge-Ampére equation:

4²u + det D²u = f; in

u = gon @ (3.71)

4u = on @

where is a bound domain in the R² with a smooth boundary @ , f is a given function

3.1 Linearization Regularization Monge-Ampére equation

the function of regularization Monge-Ampére equation

M A [u] = 4²u + det D²u (3.72)

variation of MA [u] is

D_uM A [u] = (D_yyu) (D_xx) + (D_xxu) (D_yy) 2 (D_xyu) (D_xy) 4²

= O cof(D²u)O 4² (3.73)

the linearization regularization Monge-Ampére equation

4²u +O cof(D²u)Ou = f (3.74)

36

3.3 Non-linear iteration 37

3.2 Variation formulation

the equivalent variational problem

Z

4²u vdx +

Z O cof(D²u)Ou vdx =Z

f vdxin (3.75) the weak formulation of the biharmonic term R

4²u vdxand 4u = the weak formulation of the fully non-linear term

Z

So the equivalent variational problem of equation (3:71) is Z

For non-linear problem, we usually use iterative method such as xed-point iteration, Newton's iteration etc.. Iteration method can be classi ed by the rate of convergence, q-quadratically, q-superlinearly, and q-linearly.

3.3 Non-linear iteration 38

3.3.1 Fixed-Point Iteration

Many non-linear equation are naturally formulated as xed-point problem

x = K (x) (3.79)

where K, the xed-point map may be non-linear. A solution ^x of (3:79) called a xed point of the map K. The xed-point iteration is given by

x_n+1 = K (x_n) (3.80)

This iterative method is also called non-linear Richardson iteration, Picard iteration, or the method of successive substitution.

3.3.2 Newton's method

The Newton's iteration is

x_n+1 = x_n F⁰(x_n) ¹F (x_n) (3.81)

sometimes the F⁰(x_n) ¹is not easy to nd, then we can consider use approximate the term, such as chord method, Shamanskii method or secant method etc..

3.3.3 Non-linear iteration of regularization Monge-Ampére equation

The regularization Monge-Ampére equation

F [u] = f + 4²u det D²u (3.82)

the DuF [u]is

D_uF [u] = O cof(D²u)O + 4² (3.83)

3.3 Non-linear iteration 39

Newton's iteration of F

uⁿ⁺¹ = uⁿ D_uF [uⁿ] ¹F [uⁿ] (3.84)

3.4 Basis function of BCIZ element 40

Fig. 3.12. BCIZ truangular element

3.4 Basis function of BCIZ element

To build the necessary technical tools, we shall derive and present a detailed study of the linearization of the elliptic regularization Monge-Ampere equation and its BCIZ nite el-ement approximation. Introduction to the BCIZ elel-ement, BCIZ elel-ement is conforming element, it can calculus the curvature easily, and its approximation is very well. But the basic BCIZ element has a problem, if the mesh is non-uniform mesh, then the numeri-cal result is lost the accuracy. So many people propose the revise BCIZ element such that numerical result has good approximation on non-uniform mesh.

BCIZ element:

Let P = P3. Let N = fN1; N₂; :::; N₉g

In the free-form design problem, we must to consider that the rst differential of the solution, so we choose BCIZ nite element approximation. It can easy the calculus the rst differential and curve of each element.

3.4 Basis function of BCIZ element 41

The visible degree of freedom of the the element collected in v are

v^T = w_{1 x1 y1} w_{2 x2 y2} w_{3 x3 y3} (3.85)

where the _xand _xis consider rotation, that is different from u, under Cartesian coordinate u = w; u_y = _x; u_x = _y:

'₁ = ²₁(3 2 ₁) + 2 _{1 2 3} '₂ = ²₁(y_{12 2}+ y_{13 3}) 1

2(y₁₂+ y₁₃) _{1 2 3} '₃ = ²₁(x_{12 2}+ x_{13 3}) + 1

2(x₁₂+ x₁₃) _{1 2 3} '₄ = ²₂(3 2 ₂) + 2 _{1 2 3}

'₅ = ²₂(y_{23 3}+ y_{21 1}) 1

2(y₂₃+ y₂₁) _{1 2 3} '₆ = ²₂(x_{23 3}+ x_{21 1}) + 1

2(x₂₃+ x₂₁) _{1 2 3} '₇ = ²₃(3 2 ₃) + 2 _{1 2 3}

'₈ = ²₃(y_{31 1}+ y_{32 2}) 1

2(y₃₁+ y₃₂) _{1 2 3} '₉ = ²₁(x_{31 1}+ x_{32 2}) + 1

2(x₃₁+ x₃₂) _{1 2 3} (3.86) where _i are triangular coordinate.

3.4 Basis function of BCIZ element 42

Let

= '₁ '₂ '₃ '₄ '₅ '₆ '₇ '₈ '₉ (3.87)

and

w ( ₁; ₂; ₃) = v (3.88)

Derive the element matrix Derive the element matrix of the variation equation (3:78) with BCIZ element

3.4.1 The linearization of non-linear term and element matrix

Change coordinate from the global coordinate to the local coordinate, the relationship of O^x;yandO 1; ₂; ₃ is

the w use BCIZ element to approximation, w = v

2

3.4 Basis function of BCIZ element 43

so theOv = ZBv; then the element matrix of R

(cof (D²u)Ou) Ovdx is

3.4 Basis function of BCIZ element 44

The second derivative of a function w ( ₁; ₂; ₃)with respect to x or y from (2:59) and application of the chain rule:

@²w which matrix form is

2

3.4 Basis function of BCIZ element 45

the w use BCIZ element to approximation, w = v 2

3.4 Basis function of BCIZ element 46

In this thesis, we will follow this algorithm 1. Given a initial u₀, and tolerance T

2. Fixed

3. Newton's iteration if kuⁿ⁺¹ uⁿk T then it converge, if not the iteration is diverge

4. if h² where h is mesh size then out the algorithm

5. let = =c where c is a constant, then go to 2

Chapter 4 Numerical Study

The numerical result will be given, These are three part of this chapter: Poisson equation ,biharmonic equation and Monge-Ampére equation.

4.1 Poisson Equation

Poisson Equation:

4u = f in uj^@ = g

where f and g are obtained from a given analytical solution u. We use Linear element and BCIZ element to approximate the Poisson equation. We compute the Poisson equation on different mesh size. Our calculation domain is [0; 1] [0; 1] :The boudary condition are Dirchlet type.

47

4.1 Poisson Equation 48

4.1.1 Example:

The analytical solution u = e^x+y; f = 2e^x+y and g = e^x+y: we use linear element to approximate the Poisson equation in this case.

Computed solution uh Error

h ku⁰ u_hk₁ ku⁰ u_hkL² ku⁰ u_hkH¹

1=10 5.97E-03 8.28E-04 2.04E-01 1=20 1.98E-03 2.21E-04 1.02E-01 1=40 6.28E-04 5.68E-05 5.10E-02 1=80 1.91E-04 1.44E-05 2.55E-02 1=160 5.65E-05 3.61E-06 1.28E-02

Table 1. Change of ku u^hk w.r.t. h

The convergence rate of L² norm is second order and H¹ norm is rst order. This result is same as error estimates of the biharmonic equation using BCIZ element approxi-mation.

4.1 Poisson Equation 49

4.1.2 Example:

The analytical solution u = sin (2 x) sin (2 y) ; f = 8 ²sin (2 x) sin (2 y)and g = 0:

we use BCIZ element to approximate the Poisson equation in this case.

Computed solution uh Error

h ku⁰ u_hk₁ ku⁰ u_hkL² ku⁰ u_hkH¹

2 ³ 1.23E-3 8.03E-4 4.37E-2

2 ⁴ 1.67E-4 7.54E-5 4.19E-3

2 ⁵ 1.32E-5 7.24E-6 5.32E-4

2 ⁶ 8.83E-7 7.85E-7 1.18E-4

Table 2. Change of ku uhk w.r.t. h

The convergence rate of L² norm is third order and H¹ norm is second order. This result is same as error estimates of the biharmonic equation using BCIZ element approxi-mation.

4.2 Biharmonic Equation 50

4.2 Biharmonic Equation

Biharmonic Equation:

4²u = f in uj^@ = g Ou nj^@ = h

where f, g and h are obtained from a given analytical solution u. We use BCIZ element to approximate the Biharmonic equation. We compute the Biharmonic equation on different mesh size. Our calculation domain is [0; 1] [0; 1] : The boudary condition are Dirichlet type and Neumann type. Because of the Biharmonic equation is fourth order equation, so the approximation of linear element maybe not have a high accuracy.

4.2 Biharmonic Equation 51

4.2.1 Example:

The analytical solution u = x cos(x)e^y; f = 0,g = x cos(x)e^yandOu = cos(x)e^y x sin(x)e^y x cos(x)e^y :

Computed solution uh Error

h ku⁰ u_hk₁ ku⁰ u_hkL² ku⁰ u_hkH²

2 ² 0.002971694 0.001206162 0.499530706 2 ³ 0.00065049 0.000273676 0.242128734 2 ⁴ 0.000155176 6.32838E-05 0.119339924 2 ⁵ 3.78495E-05 1.51706E-05 0.059270777 2 ⁶ 9.32505E-06 3.71417E-06 0.02954046 2 ⁷ 2.31532E-06 9.1924E-07 0.014747201

Table 3. Change of ku u^hk w.r.t. h

The convergence rate of L² norm is second order and H² norm is rst order. This result is same as error estimates of the biharmonic equation using BCIZ element approxi-mation.

4.2 Biharmonic Equation 52

4.2.2 Example:

The analytical solution u = (cos(2 x) 1) (y² 2y³ + y⁴) ; f = 0,g = x cos(x)e^y and h = 0:

Computed solution uh Error

h ku⁰ u_hk₁ ku⁰ u_hkL² ku⁰ u_hkH²

2 ² 0.01171965 0.004154116 0.645238157 2 ³ 0.004269365 0.001568274 0.307797116 2 ⁴ 0.001169401 0.000445123 0.150571686 2 ⁵ 0.000302368 0.000116688 0.074556331 2 ⁶ 7.67083E-05 2.97733E-05 0.037111559 2 ⁷ 1.93091E-05 7.5141E-06 0.018516197

Table 4. Change of ku u^hk w.r.t. h

The convergence rate of L² norm is second order and H² norm is rst order. This result is same as error estimates of the biharmonic equation using BCIZ element approxi-mation.

4.3 Monge-Ampére 53

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

References

[1] L. A. Caffarelli and X. Cabre, “Fully nonlinear elliptic equations,” volume 43 of Amer-ican Mathematical Society Colloquium Publications. AmerAmer-ican Mathematical Society, Providence, RI, 1995.

[2] L. A. Caffarelli and M. Milman, “Monge Ampere Equation: Applications to Geome-try and Optimization, Contemporary Mathematics,” American Mathematical Society, Providence, RI, 1999.

[3] W. H. Fleming and H. M. Soner, “Controlled Markov processes and viscosity solutions,”

volume 25 of Stochastic Modelling and Applied Probability. Springer, New York, sec-ond edition, 2006.

[4] D. Gilbarg and N. S. Trudinger, “Elliptic Partial Dierential Equations of Second Order, Classics in Mathematics”, Springer-Verlang, Berlin, 2001. Reprint of the 1998 edition.

[5] R. J. McCann and A. M. Oberman. “Exact semi-geostrophic ows in an elliptical ocean basin,” Nonlinearity,17(5):1891{1922, 2004.

[6] A.V. Pogorelov, "Extrinsic geometry of convex surfaces", AMS, 1973

[7] J. S. Schruben, “Formulation of a Re ector-Design Problem for a Lighting Fixture”, Journal of the Optical Society of America, V.62 N.12 (1972), 1498 1501:

[8] P. Benitez, J. C. Minano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernandez and W. Falicoff, “ Simultaneous multiple surface optical design methof in three dimen-sions”, Opt. Eng. 43(7) 1489-1502.

[9] C. E. Gutierrez, “The Monge-Ampere Equation,” volume 44, Birkhauser, Boston, MA, 2001.

[10] J. D. Benamou, B. D. Froese and A. D. Oberman, “Two Numerical Method for the Elliptic Monge-Ampere Equation”, Preprint, 2009

[11] E. J. Dean and R. Glowinshi, “Numerical solution of the two-dimensional elliptic Monge–Ampere equation with Dirichlet boundary conditions: an augmented Lagrangian approach”, C. R. Acad. Sci. Paris, Ser. I 336 (2003) 779–784

67

References 68

[12] E. J. Dean and R. Glowinshi, “Numerical solution of the two-dimensional elliptic Monge–Ampere equation with Dirichlet boundary conditions: a least-squares approach”, C. R. Acad. Sci. Paris, Ser. I 339 (2004) 887–892

[13] E. J. Dean and R. Glowinshi, “Numerical methods for fully nonlinear elliptic equations of the Monge–Ampere type”, Comput. Methods Appl. Mech. Engrg. 195 (2006) 1344–

1386

[14] P. Guan and X.J. Wang “On a Monge-Ampere equation arising in geometric optics”, J. Differential Geom, 48(1998), 205-223

[15] X. Feng and M. Neilan. “Mixed nite element methods for the fully nonlinear

Monge-Amp ere equation based on the vanishing moment method.” SIAM J. Numer. Anal.,47(2),1226-1250, 2009.

[16] X. Feng and M. Neilan. “Vanishing moment method and moment solutions for fully nonlinear second order partial di erential equations.” J. Sci. Comput., 38(1),74-98, 2009.

[17] G. P. Bazeley, Y. K. Cheung, B. M. Irons and O. C. Zienkiewicz, `Triangular ele-ments in plate bendingconforming and non-conforming solutions', Proc. Conf. on Ma-trix Methods in Structural Mechanics, WPAFB, Ohio, 1965. pp. 547-576.

[18] G. A. Mohr and A. S. Power, “Natural Cubic Element Formulation and In nite Domain Modelling for Potential Flow Problems”, ANZIAM J. 45(2003),133-143.

[19] C. A. Felippa and B. Haugen, “From the Individual Element Test to Finite Element Templates: Evolution the Patch test”, International Journal for Numerical Methods in Engineering, Vol. 38, 199-229 (1995)

[20] M. I. G. Bloor and M. J. Wilson, “An approximate analytic solution method for the biharmonic problem”, Proc. R. Soc. A (2006) 462, 1107-1121

[21] Julio Chaves, "Introduction to Nonimaging Optics", CRC, 271-324

[22] P. Benitez and J. C. Minano, “The Future of Illumination Design”, Optics and Photon-ics News, Vol. 18, Issue 5, pp. 20-25

[23] V. Oliker and E. Newman, “The Energy Conservation Equation in the Re ector Map-ping Problem”, Appl. Math. Lett. Vol. 6,No. 1,pp. 91-95, 1993

[24] Daryl L. Logan. “First Course in the Finite Element Method Using Algor”, Pws Pub-lishing, 1997.

References 69

[25] Per-Olof Persson and Gilbert Strang, "A Simple Mesh Generator in MATLAB," SIAM Review Vol. 46 (2) 2004.

[26] C. T. Kelley, “Iterative Method for Linear and nonlinear Equations”, SIAM, Philadel-phia, 1995

[27] D. Braess, “Finite element” , Cambridge University Press, 2001

[28] C. Johnson, “Numerical solution of partial differential equations by the nite element method”, Cambridge University Press, 1988

[29] Susanne C. Brenner and L. Ridgway Scott, “The Mathematical Theory of Finite Ele-ment Methods”, Springer, 2002

[30] G. Awanou “Numerical Methods for Fully Nonlinear Elliptic Equations”, International Conference on Approximation Theory, 2010

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 41-0)