1D chegaraviy qiymat masalalari uchun chekli element usullari

6.4.4 Testlashga mashq
Keling, sinov misolini ko'rib chiqaylik

Aniq yechim

Matlab kodining namunasi quyida keltirilgan:
clear all; close all; % Clear every thing so it won't mess up
% with other existing variables.

%%%%%% Generate a mesh.

x(1)=0; x(2)=0.1; x(3)=0.3; x(4)=0.333; x(5)=0.5; x(6)=0.75;x(7)=1;

U = fem1d(x);

%%%%%% Compare errors:

x2 = 0:0.05:1; k2 = length(x2);
for i=1:k2,
u_exact(i) = soln(x2(i));
u_fem(i) = fem_soln(x,U,x2(i)); % Compute FE solution at x2(i)
end

error = norm(u_fem-u_exact,inf) % Compute the infinity error

plot(x2,u_fem,':', x2,u_exact) % Solid: the exact,
% dotted: FE solution
hold; plot(x,U,'o') % Mark the solution at nodal
% points
xlabel('x'); ylabel('u(x) & u_{fem}(x)');
title('Solid line: Exact solution, Dotted line: FE solution')

figure(2); plot(x2,u_fem-u_exact); title('Error plot')
xlabel('x'); ylabel('u-u_{fem}'); title('Error Plot')

6.4 -rasmda kodni ishga tushirish orqali ishlab chiqarilgan grafiklar ko'rsatilgan. 6.4 (a) -rasmda ham aniq yechim (qattiq chiziq), ham chekli element yechimi (uzuq chiziq) ko'rsatilgan. Kichik "o" lar tugun nuqtalaridagi chekli element yechimlari qiymatlari. 6.4 - (b) rasmda bir nechta tanlangan nuqtalarda aniq va chekli element yechimlari orasidagi xatolik ko'rsatilgan (bu misolda tugun nuqtalarida nol, garchi umuman bunday bo'lmasa ham).

6.4 -(a) rasm. Aniq yechim (qattiq chiziq) va chekli elementli yechim (uzuq chiziq) va (b) ba'zi tanlangan nuqtalarda xatolik chizig'i.
Topshiriqlar

1. 
   Quyidagi ChQM ni muhokama qilamiz:
   

1. 
   Kuchsiz forma (variatsion shakl) ekanligini ko'rsating
   

bu yerda

2. 
   Chekli element yaqinlashuvi uchun tenglamalarning chiziqli sistemasini chiqaring
   

quyidagi ma’lumotlar bilan

• 
  f(x)=1;
  
• 
  tugun nuqtalar va elementlar quyidagidek indekslangan
  

• 
  basis funksiyalar bu chapkacha funksiyalari
  

va tugun nuqtalari va elementlarining tartibini o'zgartirmang; va

• 
  qattiqlik matritsasini va yuklanish vektor elementini elementlar bo'yicha yig'ing.
  

2. 
   (Bu masala drive.m, f.m va soln.m. ni o'zgartirishni o'z ichiga oladi) Yechish uchun Matlab kodidan foydalaning
   

Ikki xil to`rni sinab ko'ring: (a) drive.m da berilgan; (b) tekis to'r x<sub>i</sub> = ih, h = 1/M, i = 0, 1,. . . , M. Matlabda bajarilganda M = 10 ni oling: x= 0: 0.1: 1.
Quyidagi f(x) yoki aniq u(x) uchun masalani yechish uchun ikkita to`rdan foydalaning:

1. 
   u(x) = sin(πx) berilgan, f(x)-?
   
2. 
   given f(x) = x³, u(x)-?
   
3. 
   (qo'shimcha ball) berilgan f (x) = δ (x -1/2), bu erda δ (x) - Dirac delta funktsiyasi, u (x) nima?