Affine-weighted hat mass matrix ($W_{ij}=\int w(x){\varphi }_i(x){\varphi }_j(x)dx$)

On a one-dimensional uniform finite element mesh of spacing $h$, let $w$ be an affine function and ${\varphi }_i$ the interior piecewise-linear hat functions. The weighted mass matrix has entriesand zero entries for nonadjacent indices. The diagonal product ${\varphi }_i^2$ is symmetric about $x_i$ and integrates to $2h/3$; the neighbouring product is symmetric about $(x_i+x_{i+1})/2$ and integrates to $h/6$. The odd part of an affine function integrates to zero around each centre, giving the formulas. This evaluates a variable reaction term exactly without substituting a constant coefficient or a quadrature approximation.