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Hello forum.

I am solving the heat equation i 1d using FEM. It works fine for initial condition. However when I tried to run it for t>0 it does not work properly.

Here is my code, you can run it with this inputs and see

[Uh]=ParaFEM1D(1,0,1,10,7)

function [Uh, xnod]=ParaFEM1D(Tf,a, b, Nt, xnel)

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%% This function shows how to set up a finite element solution

%% of a two-point BVP

%% -(ku')' =f, u(a)=ua, u(b)=ub

%% using Galerkin linear Finite Elements

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%% function sol = solfem (a,b,xnel)

%% k,f,ua,ub are provided as functions in code

%% numerical integration is used in computation of stiffness matrix and rhs

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%% INPUT:

%% <a,b> = endpoints of interval

%% <xnel> = the number or location of elements

%% if xnel is a scalar, then xnel = number of elements,

%% if xnel is a vector, then it contains position of first node of each element

%% OUTPUT:

%% <sol> is the computed numerical solution at nodes

%% <xnod> position of all nodes

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

clf;

set(0,'DefaultLineLineWidth',3);

%%%%%%%%%%%%%%%%%%%%%%%%% set up grid

dt = Tf/Nt;

t = 0:dt:Tf;

if size(xnel) == 1 %% uniform grid

nels = xnel;

hel = (b-a)./nels;

xel = a : hel : b-hel;

elseif size(xnel,1) == 1

nels = size(xnel,2);

xel=xnel;

for i=1:nels-1

hel(i) = xnel(i+1)-xnel(i);

end;

else

error('wrong number of elements/grid');

end;

%% set up uniform order of elements = 1 + degree of polynomial =

%% = number of degrees of freedom

ord = zeros(nels,1) + 2; %% type of elements: linear

maxord = max(ord);

%% number of nodes

nnodes = sum(ord-1)+1;

%% derive global indexing of nodes:

%% nod(i,j) is the global number of j'th node in element i

nod = zeros(nels,maxord); myel = zeros(nnodes,2);

n = 1;

for i = 1:nels

for j = 1:ord(i)

nod(i,j) = n;

if j == 1

myel(n,2) = i;

elseif j == ord(i)

myel(n,1) = i;

else

myel(n,1) = i;

myel(n,2) = i;

end;

if j ~= ord(i)

n = n+1;

end

end;

end;

%% xnod (i=1..nnodes): coordinates of node i

xnod = zeros(nnodes,1);

for i=1:nels-1

h = xel(i+1)-xel(i);

hi = h/(ord(i)-1);

for j=1:ord(i)

xnod (nod(i,j)) = xel(i) + hi*(j-1);

end;

end;

i = nels;

h = b-xel(i);

hi=h/(ord(i)-1);

for j=1:ord(i)

xnod (nod(i,j)) = xel(i) + hi*(j-1);

end;

%%%%%%%%%%%%%%%%%%%%%%%%% set up numerical integration

%% set up quadrature parameters on the reference element (-1,1)

%% set up number of integration points nw, nodes xw, and weights w

if maxord == 1 %% exact for linears

nw = 1;

xw(1) = 0.;

w(1) = 2.;

elseif maxord == 2 %% exact for cubics

nw = 2;

xw(1) = -1/sqrt(3); xw(2) = -xw(1);

w(1) = 1; w(2) = 1;

elseif maxord == 3 %% exact for polynomials of degree 5

nw = 3;

xw(1) = -sqrt(3./5.); xw(2)=0.; xw(3) =- xw(1);

w(1) = 5./9.; w(2)=8./9.; w(3)=w(1);

end;

%%%%%%%%%%%%%%%%%%%%%%%%% matrix and rhs of linear system

b_global = sparse(nnodes,nnodes); m_global = sparse(nnodes,nnodes);

gs_global = sparse(nnodes,nnodes); rhsf = zeros(nnodes,1);

for el = 1:nels

x1 = xnod(nod(el,1)); %% left endpoint

x2 = xnod(nod(el,2)); %% right endpoint

dx = (x2-x1)/2.; %% Jacobian of transformation

%% compute element stiffness matrix and load vector

b_local = zeros(ord(el),ord(el)); %% element stiffness matrix

% gs_local = zeros(ord(el),ord(el));

m_local = zeros(ord(el),ord(el));

% gs_local = zeros(ord(el),ord(el)); %% GS = B+dt*A

f = zeros(ord(el),1); %% element load vector

%nw is the order of Gaussian Integration

for i = 1:nw

x = x1 + (1 + xw(i))*dx; %% x runs in true element,

%% xw runs in reference element

[psi,dpsi] = shape(xw(i),ord); %% calculations on ref.element

% kval = feval(@kfun,x);

% fval = feval(@rhsfun,0,x);

fval = feval(@exfun,0,x);

f = f + fval * psi * w(i)*dx;

b_local = b_local + (psi*psi')* w(i)*dx; %mass vector

m_local = m_local + w(i)*(dt*(dpsi*dpsi')/dx/dx) * dx;%stiffness matrix

% gs_local = gs_local + w(i)*(dt*(dpsi*dpsi')/dx/dx) * dx + (psi*psi')* w(i)*dx ;

end

%% add the computed element stiffness matrix and load vector to

%% the global matrix and vector

rhsf(nod(el,:)) = rhsf(nod(el,:)) + f(:); % F = load vector

b_global(nod(el,:),nod(el,:)) = b_global(nod(el,:),nod(el,:)) + b_local; % B = mass matrix

m_global(nod(el,:),nod(el,:)) = m_global(nod(el,:),nod(el,:)) + m_local; % A = stiffness matrix

% gs_global(nod(el,:),nod(el,:)) = gs_global(nod(el,:),nod(el,:)) + gs_local;

end

%% impose Dirichlet boundary conditions

xa = exfun(0,a); xb = exfun(0,b) ;

Uh = zeros(Nt,nnodes);

for i=1:nnodes

% rhsf(i) = rhsf(i) - m_global(i,1)*xa - m_global(i,nnodes)*xb ;

rhsf(i) = rhsf(i) - (dt*m_global(i,1)-b_global(i,1))*xa - ...

(dt*m_global(i,nnodes)-b_global(i,nnodes))*xb ;

m_global(i,1) =0; m_global(1,i)=0;

b_global(i,1) =0; b_global(1,i)=0;

m_global(i,nnodes)=0; m_global(nnodes,i)=0;

b_global(i,nnodes)=0; b_global(nnodes,i)=0;

end

m_global(1,1) = 1; b_global(1,1) = 1; rhsf(1,1) = xa;

m_global(nnodes,nnodes)=1;b_global(nnodes,nnodes)=1; rhsf(nnodes,1) = xb;

for n=1:Nt

fprintf('itetation %d of %d\n',n,Nt);

% k = t(n+1) - t(n); % time step;

if n == 1

Uh(n,:) = b_global\rhsf;

else

% Uh(ntime,:) = Uh(ntime-1,:)*(gs_global\(dt*b_global));

Uh(n,:) = (b_global+dt*m_global)\(b_global*Uh(n,:)'+dt*rhsf);

end

figure(1)

exfun(dt*(n-1),xnod)'

plot(xnod,exfun(dt*(n-1),xnod),'k',xnod,Uh(n,:),'r--');

pause(0.5),legend('Exact solution','Numerical solution (linear FE)');

end

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%% end of algorithm

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

function [y,dy] = shape(x,n)

%% shape function on reference element (-1,1)

%% n = 2: linear

%% n = 3: quadratic (must be coded)

if n == 2

y (1,:) = .5.*(1-x);

y (2,:) = .5.*(1+x);

dy (1,:) = -.5;

dy (2,:) = .5;

end

function y = exfun(t, x)

y= sin(pi*x).*exp(-pi^2*t) + sin(2*pi*x).*exp(-4*pi^2*t);

% dy = pi*cos(pi*x).*exp(-pi^2*t) + 2*pi*cos(2*pi*x).*exp(-4*pi^2*t);

Please help

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