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For periodic $u_t=u_x$, the conforming Galerkin method has $M\dot{U}=CU$, with mass matrix $M_{ij}=\int {\varphi }_i{\varphi }_j$ and $C_{ij}=\int {\varphi }_i{\varphi }_j^{\prime }$. Integration by parts and periodicity give $C^T=-C$. ThereforeThe conserved quantity is the squared L2 norm of the finite element function. On a uniform one-dimensional mesh of spacing $h$, the piecewise-linear hat functions give $M_{ii}=2h/3$, $M_{i,i\pm 1}=h/6$, $C_{i,i+1}=1/2$ and $C_{i,i-1}=-1/2$.

The L2 inner product is the inner product $⟨f,g⟩=\int f\overline{g}d\mu$ on L2 space. Its induced norm is the L2 norm, and the mass matrix represents it in a finite element basis.

The mass matrix is the Gram matrix of a finite element basis in the L2 inner product. Its coefficient quadratic form is the squared L2 norm of the represented trial function, so it is a positive-definite matrix for a linearly independent basis.

Let $x_m=mh$ be a uniform periodic mesh, with indices interpreted modulo the number of nodes. Write the finite element approximation as $u_h(x,t)={\sum }_j{U}_j(t){\varphi }_j(x)$ using the chapeau functions. The Galerkin method requiresThe mass matrix has $M_{mm}=2h/3$ and $M_{m,m\pm 1}=h/6$. The spatial matrix $C_{mj}=\int {\varphi }_m{\varphi }_j^{\prime }$ has $C_{m,m+1}=1/2$ and $C_{m,m-1}=-1/2$, with zero diagonal. Therefore the semidiscrete equations areor equivalently $M\dot{U}=CU$. The coefficients come from integrating the products of the two overlapping piecewise-linear hat functions on each interval; periodicity supplies the wraparound entries.

The stiffness matrix represents a variational bilinear form $a$ in a finite element basis. A symmetric coercive bilinear form gives a positive-definite matrix. Local support of the piecewise-linear hat functions produces a sparse matrix, allowing efficient numerical solution.