MCS 571 FEM Simple Example -- Fall 2001

(Partial MATLAB Notation Used)

P/EPDE: u_t = a₀*u_xx +b₀*u_x+c₀*u+s₀ on [x₁,x₃], u=u(x,t) with constant coefficients.
Subinterval Elements:
m=3 nodes (x₁,x₂,x₃);
ne=m-1=2 elements, {E₁=[x₁,x₂], E₂=[x₂,x₃]);
Step Sizes: h_i=x_i+1-x_i = h, fixed, for i=1:2.
Linear Basis: Phi = {[ (x₂-x)/h₁ (x-x₁)/h₁,0] on [x₁,x₂]; [0, (x₃-x)/h₂, (x-x₂)/h₂] on [x₂,x₃]}
"Vector" of Unknowns:
UH=Uhat=[U₂(t)]
Global Vector:
U(t)=[U₁(t);U₂(t);U₃(t)]
Boundary Conditions:
U₁(t)=u(x₁,t);
U₃(t)=u(x₃,t).
Initial Condition:
U₂=u(x₂,0);
Galerkin Approximation:
U(x,t)= U₁(t)*phi₁(x) +U₂(t)*phi₂(x) +U₃(t)*phi₃(x) =u(x₁,t)*phi₁(x) +U₂(t)*phi₂(x) +u(x₃,t)*phi₃(x).
Elemental Assembly Matrices:
A₁=[1,0,0; 0, 1,0];
A₂=[0,1,0; 0, 0,1];
Elemental Mass Matrices:
m₁=int([ phi₁(x)*phi₁(x), phi₁(x)*phi₂(x); phi₂(x)*phi₁(x), phi₂(x)*phi₂(x)], on E₁) dx =(h/6)*[2,1;1,2] = m₂
by integration and symmetry.
Global (g) Mass Matrix:
M_g = A₁'*m₁*A₁ +A₂'*m₂*A₂ =(h/6)*[2,1,0;1,4,1;0,1,2];
Checking:
M=M_1,1 = int(phi₂(x)*phi₂(x), on {E₁+E₂}) = 4*h/6 = M_g,2,2, checks;
also
M_g,2,1=int(phi₂(x)*phi₁(x), on {E₁+E₂}) = h/3 = M_g,2,3.
Stiffness Matrix:
K=K_1,1=K_g,2,2 = int(a₀*(dphi₂(x)/dx)^2 -b₀*phi₂(x)*dphi₂(x)/dx -c₀*phi₂^2(x), on {E₁+E₂}) =2*(a₀/h-h*c₀/3);
K_g,2,1 = int(a₀*(dphi₂(x)/dx)*(dphi₁(x)/dx) -b₀*phi₂(x)*dphi₁(x)/dx -c₀*phi₂(x)*phi₁(x), on {E₁+E₂}) =-a₀/h+0.5*b₀-h*c₀/3;
K_g,2,3 = int(a₀*(dphi₂(x)/dx)*(dphi₃(x)/dx) -b₀*phi₂(x)*dphi₃(x)/dx -c₀*phi₂(x)*phi₃(x), on {E₁+E₂}) =-a₀/h-0.5*b₀-h*c₀/3.
Main Forcing/Source Term:
S=S_1,1=S_g,2,2 = int(phi₂(x)*s₀, {E₁+E₂}) =s₀*h.
Galerkin FEM ODE:
M*U₂'(t)+K*U₂(t)=RHS(t)= S -M_g,2,1*u_t(x₁,t) -M_g,2,3*u_t(x₃,t) -K_g,2,1*u(x₁,t) -K_g,2,3*u(x₃,t)
Solutions:
PPDE:
U₂(t)=exp(-(K/M)*t)*(u(x₂,0) +int(exp((K/M)*q)*RHS(q), q on [0,t])/M;
the integral depending on the variable B.C.s.
EPDE (steady state: U₂ and B.C.s independent of t):
U₂=RHS/K;

Web Source: http://www.math.uic.edu/~hanson/mcs571/fem01example.html
MCS 571 HomePage: http://www.math.uic.edu/~hanson/mcs571/
Email Comments or Questions to hanson@uic.edu