 Solution

Define the mass matrix $M_{ij}={\int }_0^1{\varphi }_j{\varphi }_i$ and stiffness matrix $K_{ij}={\int }_0^1{\varphi }_j^{\prime }{\varphi }_i^{\prime }$ for $1\le i,j\le N$. With coefficient vectors $U^m$, the step system isThe matrices are symmetric, and $V^T\mathsf{A}V=\Vert v_h{\Vert }_2^2+\Delta t(\Vert v_h^{\prime }{\Vert }_2^2+\Vert v_h{\Vert }_2^2)>0$ for nonzero $V$, proving positive definiteness.

For $h=1/2$, the first basis function is the full tent at $1/2$, and the last is the half tent ending at $1$. Integration givesSince $\Delta t=2/2=1$, the requested matrix in this normalization isHere the matrix symbol $M$ is distinct from the printed integer number of time steps.