This is MATLAB code for (ut=uxx ) on {0,1} where
u(x,t)=sin(2pi*x)exp[-(2 pi).^2]*t,
I.C: u(x,t) = sin(2pi*x);
B.C: u(0,t) = u(1,t) =0;
L=1; n=1;
t_final =0.1;
N = 32; %elements
dx =1/N;
dt = 0.01;
x = 0:dx:L;
t=0:dt:t_final;
for i=1:length(t)
y=sin(2*pi*x)*(exp((-(2*pi).^2)*t(i)));
y(1)=0;
y(end)=0;
figure(1)
plot(x,y)
hold on
end
%% Backward Euler for FEM solution
% L = 2;
xx =0:dx:L;
nx = length(xx); %Nodes
M = spdiags([(dx/6)*ones(nx,1), (2*dx/3)*ones(nx,1), (dx/6)*ones(nx,1)],[-1, 0,1],nx,nx) ; %% Mass matrix
A = spdiags([(-1/dx)*ones(nx,1), (2/dx)*ones(nx,1), (-1/dx)*ones(nx,1)],[-1, 0, 1],nx,nx) ;%% Stiffness matrix
delta_t = 0.001;
tspan = 0:delta_t:t_final;
yy = zeros(1,nx);
for nt=1:length(tspan)
yy = ( sin(2*pi*xx/L)*exp((-(2*pi).^2)*tspan(nt)))*(M/(M+delta_t*A));
yy(1)= sin(2*pi*xx(1));
yy(end)=0;
figure(2)
plot(xx,yy)
hold on
end
I used formula for error, error = norm(exact solution- FEM solution)
but discretization error does not converges to zero.
I obtained following errors by changing no of elements;
N Error
4 0.00100000
8 0.00074918
16 0.0007419
32 0.00073473
64 0.00073303
128 0.00073274
256 0.00073274
512 0.00073470
Saad Najeeb sir, I also want it should converge to zero but if you please see the code, where is problem or discretization error formula is not doing well?
There are many sources for error in FEM, not only discretization error like:
1. Formulation error: It results from the use of elements that do not describe the behavior of the physical problem.
2. Numerical error: It includes truncation errors or round-off errors. A slight error due to round-off in one element is likely to result in appreciable error due to cumulative effect of these errors throughout the domain.
3. Human error: It results due to taking the wrong input data.
also look at
https://www.simscale.com/blog/2016/06/errors-in-fea-and-singularities/
https://www.accessengineeringlibrary.com/browse/introduction-to-the-finite-element-method-third-edition/c9780072466850ch14lev1sec05
good luck
Mr., Khaled Megahed.. yes, all points in your answer may be cause errors.
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