# Stability condition for explicit time FEM for parabolic pdes

If we discretize a parabolic pde to obtain the system of ODE's $$\frac{\boldsymbol{B}}{\Delta t} \boldsymbol{u}_k = (\boldsymbol{K} + \frac{\boldsymbol{B}}{\Delta t}) \boldsymbol{u}_{k-1} + \boldsymbol{f}_k$$ where $$\boldsymbol{B}$$ is the mass while $$\boldsymbol{K}$$ is the stiffness matrix, is there any condition on $$\frac{\boldsymbol{B}}{\Delta t}$$ such that this system is stable?

This system results from discretizing a parabolic PDE via forward Euler.

• You are missing the $B$ matrix on the right hand side of your equation. – Wolfgang Bangerth Feb 7 '20 at 4:51
• Yes, thanks. It's fixed now! – secondrate Feb 8 '20 at 15:48

• Hi @Wolfgang, thanks so much for your help. There are 2 things I'd like to ask: 1) In your notes, you write that the mass matrix $M \approx h^d I$ where $h$ is the discretization length in the spatial domain. Any references (i.e. books) for more details on this? 2) What about a CFL-type condition for FEM? Must there be a relationship between $\|M\|_2$ and $\Delta t$? Any references would be appreciated. – secondrate Feb 8 '20 at 15:54
• @secondrate: About the relationship of the mass matrix and $h^dI$: Think of the relationship of any matrix $K$ to an operator $\cal K$ as $K_{ij}=(\varphi_i, {\cal K} \varphi_j)$. Then it's clear that the operator that $M$ corresponds to is simply the identity operator. The $h^d$ comes out of the integration: $\varphi_i$ and $\varphi_j$ only overlap on the set of cells adjacent to one vertex, and the total integration area/volume where the product is nonzero is then of order $h^d$ in $d$ dimensions. – Wolfgang Bangerth Feb 9 '20 at 17:46
• I don't understand your second question, though. Are you asking whether the 2-norm of $M$ somehow enters the CFL limit for $\Delta t$? The answer is no. – Wolfgang Bangerth Feb 9 '20 at 17:48