Past exam of the approximation theory and finite element methods course of the University of Oxford / 2008 / B / 8 / c

Write $u_h^m={\sum }_i{U}_i^m{\phi }_i$ and define the mass matrix and reaction-diffusion stiffness matrixThen each step isBoth component matrices are symmetric, so $A$ is symmetric. For a nonzero coefficient vector $v$, its finite element function $v_h$ is nonzero andThus $A$ is also positive definite and every update has a unique solution. Because nonneighboring hats have disjoint interior supports, the matrix is tridiagonal in nodal order.