# Norm of operator in finite element discretization of Heat equation

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.

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 $$\Delta t$$.
• Hi, thanks for the reply. I am confused about 2 things: 1) I'm not interested in showing stability but rather I want to show that $\|A\|_2 < 1$. Does the approach you mentioned above work for this or is it only to show that the spectral radius is less than 1? 2) What do you mean by express $u_k,u_{k+1}$ as eigenmodes? I have never heard of the method you showed above before. Do you have a reference for this? May 1 '20 at 1:49
• 1. Why do you want to show that $||A||_2 < 1$? --- 2. $u_k = \sum_{j = 1}^{n} \mu_k^j w_j$. Using the orthogonality among the eigenvectors $(w_i)$, we get an expression between $\mu_{k+1}^{j}$ and $\mu_{k}^{j}$. May 1 '20 at 1:56
• Note also $|| ( I - M^{-1} K )^{-1} x ||_2 = || ( M^{-1/2} ( I - M^{-1/2} K M^{-1/2} ) M^{+1/2} )^{-1} x ||_2 = || M^{-1/2} ( I - M^{-1/2} K M^{-1/2} )^{-1} M^{+1/2} x ||_2 = || ( I - M^{-1/2} K M^{-1/2} )^{-1} M^{+1/2} x ||_{M^{-1}}$ May 1 '20 at 2:00
• I'm using it for error estimates. How does the above help in finding the 2-norm of $I-\Delta t M^{-1}K$? May 1 '20 at 2:08
• When you use the expression with $M^{-1}$-norm, the operator $I - \Delta t M^{-1/2} K M^{-1/2}$ is now symmetric and its eigenvectors are the ones of $(K, M)$. So studying $|| ( I - \Delta t M^{-1} K )^{-1} x ||_2 / || x ||_{2}$ is the same as studying $|| ( I - \Delta t M^{-1/2} K M^{-1/2})^{-1} y ||_{M^{-1}} / || y ||_{M^{-1}}$ (where $y = M^{1/2} x$) May 1 '20 at 2:28