数学Midterm代考 期中代考 Math 574代写

Math 574

Midterm # 2: due April 19

 

数学Midterm代考 Let (xi, yj)0≤i,j≤N be uniform grids discretize the domain Ω and Let Uij de- note the approximation of u at point (xi, yj) and Fij = f (xi, yj).

Problem 1. Consider finite difference method to solve the Laplace prob- lem

−uxx − uyy = sin(πx) sin(πy) in Ω = (0, 1)2,

and

u = 0 on the boundary ∂Ω.

Let (x_i, y_j)_0≤i,j≤N be uniform grids discretize the domain Ω and Let U_ij de- note the approximation of u at point (x_i, y_j) and F_ij = f (x_i, y_j).

a)Writedown the finite difference method for solving the Laplace equation above.

b)In MATLAB implementation, we use the command

U = zeros(N+1, N+1);   数学Midterm代考

to generate an initial guess of (U ⁰ )_0≤i,j≤N to approximate u(x_i, y_j). To solve for U_ij satisfies part a), write down a Gauss-Seidel iteration to generate a

U^(k_ij≈ Uij .

c)In MATLAB code, we use the command

f = @(x,y) sin(pi*x).*sin(pi*y);

to define the function f (x, y) = sin(πx) sin(πy) and use the command

[X,Y] =  meshgrid(0:h:1); F = f(X,Y);

to generate F_ij = f (x_i, y_j). Implement the Gauss Seidel iteration in part b). Set the tolerance tol = 1e-9, and

stop the iteration if err < tol. To plot the approximation U , use the com- mand

surf(X,Y,U)

d)For N = 8, 16, 32, record the number of the GS iteration in part c) and theerror   数学Midterm代考

Note that the exact solution u(x, y) = ¹ sin(πx) sin(πy).

e)Now suppose that the error e_h≈ Ch^α for some constants C and α. Then

How do the values of α computed in part (d) compare to the rates of con- vergence for these norms predicted by the theory? Make sure to hand in a copy of your program as m file and the values you computed of e_1/8, e_1/16, e_1/32 and α.

Problem 2. a) Write a computer program to approximate the solution of the boundary value problem:   数学Midterm代考

−u” + u = 1, 0 < x < 1, u(0) = 0 u(1) = 1.

by the finite element method using continuous piecewise linear elements on a uniform mesh of width h.  Use h = 1/100 and 1/200.

b)Let u_I(x) denote the piecewise linear interpolant of u, e., the piecewise linear function satisfying

u_I(x_i) = u(x_i), i = 0, · · · , N, and e(x) = u_I(x) − u_h(x).

Let

Have  the  computer  determine  E⁽¹⁾,  E⁽²⁾,  E⁽³⁾,  for  h  =  1/100  and  1/200.

Note: the true solution of the boundary value problem is given by

c)Now suppose that the error E^(k)≈ C_kh^αk for some constants C_k andα_k.The error Eh gives an approximation to “u − u_h“_L2(0,1)  and the errorEh gives an approximation to “∇(u − u_h)“_L2(0,1).  How do the values of α₂and α₃ computed in part (c) compare to the rates of convergence for thesenorms predicted by the theory? Make sure to hand in a copy of your program as m file and the values you computed of E_1/100, E_1/200, and α.

 

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