Dirichlet discrete Laplacian

On $J$ interior points, the one-dimensional centered second-difference matrix with homogeneous Dirichlet boundaries has eigenvalues

For $-\Delta +I$ on the unit square with zero Dirichlet boundary conditions, the continuous eigenvalues are ${\lambda }_{m,n}=1+{\pi }^2(m^2+n^2)$, whereas the mesh-$1/N$ five-point Laplacian gives ${\Lambda }_{m,n}=1+4N^2[{\sin }^2(m\pi /(2N))+{\sin }^2(n\pi /(2N))]$. The inequality $\sin z\le z$ yields ${\Lambda }_{m,n}\le {\lambda }_{m,n}$. For fixed indices, Taylor's theorem gives an error of order $N^{-2}$; for indices $m=n=N-1$, division by $N^2$ gives the positive limit $2{\pi }^2-8$. Thus an error estimate for fixed smooth eigenfunctions need not be uniform over all grid-resolvable eigenfunctions.

If $L$ is the negative semidefinite Dirichlet discrete Laplacian, the Crank-Nicolson amplification matrixhas eigenvalues $(1+\mu {\lambda }_j/2)/(1-\mu {\lambda }_j/2)$ of modulus at most one for every $\mu \ge 0$.

If $T$ is the one-dimensional Dirichlet second-difference matrix on $m$ interior points, the two directional parts of the two-dimensional five-point Laplacian areThey commute, and $A_x+A_y$ is the Kronecker-sum discretization of the Laplacian.

The split stephas amplification matrixOn a common directional eigenvector its eigenvalue is $(1+\mu {\lambda }_p)/(1-\mu {\lambda }_q)$.

On the $m$ by $m$ Dirichlet grid with $h=1/(m+1)$, the exact discrete stability condition for $m>1$ isThe mesh-independent sufficient condition is $0<\mu \le 1/2$, and the exact bound tends to $1/2$ as $h\to 0$.

Adding the correctiongivesIts common-basis eigenvalues areSince ${\lambda }_p,{\lambda }_q<0$, one has $0<d_{pq}\le 1$ for every $\mu >0$.

On a three-dimensional Cartesian grid, the seven-point Dirichlet Laplacian adds the six nearest-neighbour values and subtracts six times the central value. Extending a grid vector by zero on the boundary gives the discrete energy identityso its matrix is symmetric negative definite.