A FEniCS Tutorial - FEniCS Project

if we identify the matrix M with entries M ij as above and the matrix K with
entries
∫
K ij = ∇ˆφ i ·∇φ j dx. (77)
Ω
The matrix M is often called the ”mass matrix” while ”stiffness matrix” is a
common nickname for K. The associated bilinear forms for these matrices, as
we need them for the assembly process in a FEniCS program, become
∫
a K (u,v) = ∇u·∇vdx, (78)
Ω
∫
a M (u,v) = uvdx. (79)
Ω
The linear system at each time level, written as AU k = b, can now be
computed by first computing M and K, and then forming A = M + ∆tK at
t = 0, while b is computed as b = MU k−1 +∆tMF k at each time level.
The following modifications are needed in the d1_d2D.py program from the
previous section in order to implement the new strategy of avoiding assembly
at each time level:
• Define separate forms a M and a K
• Assemble a M to M and a K to K
• Compute A = M +∆tK
• Define f as an Expression
• Interpolate the formula for f to a finite element function F k
• Compute b = MU k−1 +∆tMF k
The relevant code segments become
# 1.
a_K = inner(nabla_grad(u), nabla_grad(v))*dx
a_M = u*v*dx
# 2. and 3.
M = assemble(a_M)
K = assemble(a_K)
A = M + dt*K
# 4.
f = Expression(’beta - 2 - 2*alpha’, beta=beta, alpha=alpha)
# 5. and 6.
while t