The stiffness matrix represents a variational bilinear form 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.
For conforming piecewise affine elements on uniform squares split along their positive diagonals, a lower-triangle element matrix is in consecutive vertex order. Finite element assembly gives diagonal entry 4 and four axis-neighbor entries . The diagonal mesh edges contribute zero coupling because the corresponding local shape-function gradients are perpendicular. Strong Dirichlet elimination removes known-neighbor columns and adjusts the load.
For with homogeneous Dirichlet boundary conditions, the conforming piecewise-linear hat functions on a uniform grid give a diffusion stiffness matrix and a mass matrix . Their nonzero entries are , , , and , with symmetric counterparts. The sum is a positive-definite matrix because its coefficient quadratic form is . For forcing , the load at is .
For with homogeneous mixed Dirichlet and Neumann boundary conditions, testing the Backward Euler method finite element equation with the new solution gives the displayed identity. Both the squared derivative norm and the squared function norm are required in the dissipation, because diffusion and reaction contribute separately. Replacing the full norm by its seminorm omits the reaction term. Positivity gives unconditional decay of the norm.
