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How to solve for 1D non homogenous ODE by Finite element method

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MUTHULINGAM
MUTHULINGAM 2024-9-10，10:01
编辑： Torsten 2024-9-10，10:11
Ran in:
I am unable to input the dirichlet condition into this non homogenous ode. Please point out my mistake.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%% -u" + u = - 5x + 4 0 < x <= 1 %%%%
%%%% u(0) = 0, u(1) = 2 %%%%
%%%% Exact solution u = (exp(x)*(3*exp(1) + 4))/(exp(2) - 1) - 5*x - (exp(-x)*(3*exp(1) + 4*exp(2)))/(exp(2) - 1) + 4 %%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
clear all;
close all;
format long;
N = 20;
x0 = 0;
xN = 1;
h = 1/N;
for j = 1:N+1
X(j) = x0 + (j-1)*h;
end
f = @(x)(- 5*x +4);
A = zeros(N+1,N+1);
F = zeros(N+1,1);
%%%% Local stiffness matrix %%%%
a = [1/h -1/h ; -1/h 1/h] ;
for i=1:N
phi1 = @(x)(X(i+1)-x)/h; %%%% linear basis function %%%%
phi2 = @(x)(x-X(i))/h; %%%% linear basis function %%%%
f1 = @(x)f(x)*phi1(x); %%%% integrand for load vector %%%%
f2 = @(x)f(x)*phi2(x); %%%% integrand for load vector %%%%
v(1,i) = gauss(f1,X(i),X(i+1),1); %%%% element wise values of
v(2,i) = gauss(f2,X(i),X(i+1),1); %%%% load vector
end
Unrecognized function or variable 'gauss'.
%%%% Assembling %%%%
for i=1:N
A([i i+1],[i i+1]) = A([i i+1],[i i+1]) + a;
F([i i+1],1) = F([i i+1],1) + v([1 2],i);
end
%%%% Dirichlet Boundary condition %%%%
% F(N+1,1) = F(N+1,1)+2;
fullnodes = [1:N+1];
%%%%% Dirichlet boundary condition %%%%%
freenodes=setdiff(fullnodes,[1]);
Uh = zeros(N+1,1);
Uh(N+1,1) = 2;
%%%% Approximate solution %%%%
Uh(freenodes)=A(freenodes,freenodes)\F(freenodes,1);
%%%% Exact solution %%%%
U = zeros(N+1,1);
for i =1:N+1
U(i) = (exp(X(i))*(3*exp(1) + 4))/(exp(2) - 1) - 5*X(i) - (exp(-X(i))*(3*exp(1) + 4*exp(2)))/(exp(2) - 1) + 4;
end
error = max(abs(U-Uh));
[U Uh]
plot(X,U,X,Uh,'o')

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回答（1 个）

Torsten
Torsten 2024-9-10，10:09
编辑：Torsten 2024-9-10，10:11
I usually set
A(1,1) = 1, F(1) = 0, A(N+1,N+1) = 1, F(N+1) = 2
and only loop from 2 to N here:
%%%% Assembling %%%%
for i=1:N
A([i i+1],[i i+1]) = A([i i+1],[i i+1]) + a;
F([i i+1],1) = F([i i+1],1) + v([1 2],i);
end
I don't know if this is "Finite-Element-like".

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