I am solving the heat equation discretized spatially via FEM and temporally via backward Euler. I get the system $$M \dot{u} = K u +f$$ where $u$ is a vector representing the solution at spatial locations.
For piecewise linear basis functions, the mass matrix is defined as
while the stiffness matrix $K$ is
By discretizing this in time, I get $$u_{k+1} = Au_k + By_k$$ for some matrix $B.$ Here, $A = (I-\Delta t M^{-1}K)^{-1}$ where $\Delta t$ is the time step size.
Is it possible to show that $$\|A\|_2 < 1$$ for $\Delta t$ sufficiently small? I would like an analytical solution without computing this in Matlab.
I have a hard time analyzing this because $M^{-1}K$ is not symmetric although $M,K$ are symmetric. Otherwise, I could have just used the fact that the spectral radius and the 2 norm coincide.