On the page 99 in the Chapter 5.2 of Prof. Endre Süli's lecture notes on FEM for PDEs (see: https://people.maths.ox.ac.uk/suli/fem.pdf), he derived the mass matrix for forward Euler scheme and the stiffness matrix for backward Euler scheme for solving 1-d heat equation. And he said "It is a simple matter to show that this matrix is tri-diagonal and has the form ..." However, it seems not a simple matter to me. Could anyone read the page 98-99, and then provide me with a step-by-step illustration about how to derive the mass matrix and the stiffness matrix, and finally how to represent the evolution problem in a linear system like $\mathbf{A} \vec{U} = \vec{F}$?
2026-02-24 11:58:14.1771934294
The mass matrix and the stiffness matrix in finite element method for heat equation
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