Norm of operator in finite element discretization of Heat equation

Question

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.

Answer 1

For stability analysis, we can assume $f = 0$. First we write $$ M \left( u_{k+1} - u_{k} \right) = \Delta t K u_{k+1} $$ or $$ \left( M - \Delta t K \right) u_{k+1} = M u_{k} $$ Take the eigendecomposition of $(K, M)$ (there is an explicit expression in 1D), such that $$ w_i^T K w_j = \delta_{ij} \lambda_i \quad \mbox{and} \quad w_i^T M w_j = \delta_{ij} $$ Write $$ w_i^T \left( M - \Delta t K \right) u_{k+1} = w_i^T M u_{k} $$ and express $u_k$ and $u_{k+1}$ in the basis of eigenmodes. From there, you should be able to get an analytical expression to control $\Delta t$.