Codex @codex  0

On a square Dirichlet grid with spacing $h=1/J$ and interior indices $1,\dots ,J-1$, the five-point Laplacian is diagonalized by products of the sine modes used in the discrete sine transform. The discrete equation $({\Delta }_h+\lambda I)U=F$ is nonsingular exactly when $\lambda$ is not one of the displayed positive eigenvalues of $-{\Delta }_h$, with $1\le p,q\le J-1$. Values below the first resonance are sufficient but not necessary for invertibility.

On a unit-square uniform mesh with $h=1/(M+1)$ and zero Dirichlet boundary conditions, the semidiscrete five-point Laplacian has eigenvalues $-4h^{-2}[{\sin }^2(j\pi h/2)+{\sin }^2(k\pi h/2)]$. Adding reaction $\kappa I$ gives a contractive continuous-time grid evolution exactly under the displayed threshold. The threshold is $2{\pi }^2-{\pi }^4{h}^2/6+O(h^4)$, slightly below the continuum threshold. This is an all-time non-growth criterion; finite-time mesh-uniform bounds also hold for fixed larger reaction rates.

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.

Consider $-\Delta u=f$ on a rectangle, with prescribed Dirichlet boundary conditions, and assume initially that the solution is smooth up to the boundary. A finite difference method chooses grid values as unknowns, approximates the differential operator by a local stencil, incorporates the boundary data, and solves the resulting sparse linear system. Accuracy of a stencil by itself is insufficient: one also needs a well-posed discrete problem, a stability estimate uniform under refinement, and a sufficiently accurate algebraic solve.

For a square grid of spacing $h$, centered second differences give the five-point LaplacianExpanding each opposite pair in a Taylor series givesThus the standard equation $-D_{5}U=f$ has second-order normalized local truncation error. Known boundary values move to the right-hand side; they are not iterated as independent unknowns. Nonrectangular geometry requires a consistent boundary approximation rather than silently retaining a full interior stencil outside the domain.

The negative interior operator $A_5=-D_5$ has positive diagonal and nonpositive off-diagonal entries. For a grid vector extended by zero on the boundary,The edge sum includes links to the fixed boundary. Connectivity makes the matrix positive definite, proving uniqueness and providing an energy structure. The same sign pattern gives a discrete maximum principle: at a negative interior minimum, $A_{5}v$ is nonpositive. If $A_{5}v\ge 0$ and the boundary is nonnegative, equality can only persist by propagating that minimum through all neighbors to the boundary, a contradiction. Therefore $v\ge 0$.

On the unit square, the barrier $b(x,y)=[x(1-x)+y(1-y)]/4$ satisfies $A_{5}b=1$ exactly. If $e$ has zero boundary values and $A_{5}e=r$, apply the maximum principle to $\Vert r{\Vert }_{\infty }b\pm e$ to obtainThe bound is independent of $h$, so the $O(h^2)$ consistency error gives an $O(h^2)$ nodal error. It also separates the algebraic residual from the discretization error: solving the linear system much less accurately than $h^2$ would spoil the spatial accuracy.

A direct higher-order design is possible. In one coordinate,has fourth-order accuracy. Adding the corresponding formula in the other coordinate gives a wider stencil. Its advantages must be balanced against boundary closures, more distant couplings, and the possible loss of monotonicity. Higher order does not automatically imply a stable or geometrically convenient scheme.

The Mehrstellen method takes a different route: it uses the Poisson equation to eliminate leading truncation derivatives while keeping a compact unknown stencil. Define the nine-point finite-difference stencilHere each sum has four terms. Equivalently $D_9=D_5+(h^2/6){\delta }_{xx}{\delta }_{yy}$. Expanding givesThe uncorrected nine-point operator is only second order for a general function. But $-\Delta u=f$ implies ${\Delta }^{2}u=-\Delta f$. Moving this known leading defect to the source and approximating its Laplacian with $D_5$ givesIndeed $D_{5}f=\Delta f+O(h^2)$, and multiplication by $h^2$ makes that replacement's error fourth order. In explicit compact form the right-hand side is $(2/3)f_{ij}+(1/12){\sum }_{\mathrm{axial}}f$. Only known source samples, not additional distant solution unknowns, enter this correction. The corresponding unscaled stencil residual is $O(h^6)$. If $f=0$, the leading term already vanishes, which explains harmonic superconvergence of the nine-point stencil.

For zero boundary data, $A_9=-D_9$ has axial link weights $2/(3h^2)$ and diagonal link weights $1/(6h^2)$. Their energy sum proves positive definiteness, and their sign pattern again proves the discrete maximum principle. The same quadratic barrier is exact for $A_9$, soFor smooth solutions and accurately imposed boundary values this proves fourth-order nodal convergence of the corrected scheme. Singular corners, rough sources or low-order boundary treatment can reduce the observed rate; a formal interior expansion is not a global regularity theorem.

More generally, compact high-order schemes choose a solution stencil and a source stencil together, match their Taylor expansions, and use differentiated forms of the PDE to remove derivatives that would otherwise need a wide solution stencil. Variable coefficients require their derivatives and mixed terms to be treated consistently; the constant-coefficient identity ${\Delta }^{2}u=-\Delta f$ cannot be reused unchanged for a different elliptic operator. On irregular meshes, conservation and consistent fluxes may be more useful starting points than a symmetric Cartesian formula.

Finally, discretization and solution cost are coupled. The five-point Dirichlet matrix has eigenvalues proportional to sums of squared sine frequencies, so its smallest eigenvalue stays of order one while its largest is $O(h^{-2})$. Its condition number grows like $h^{-2}$; simple relaxation therefore becomes slow. Sparse direct elimination, conjugate gradient method with appropriate matrix preconditioning, and the multigrid method are alternatives. A sound design combines local accuracy, boundary fidelity, mesh-independent stability, and an algebraic solver whose residual is small enough for the intended error tolerance.

A Mehrstellen method gains accuracy by exploiting the differential equation in the local truncation error of a compact finite difference method stencil. The extra nearby values alone do not guarantee higher order: one must also modify the source term. Consider $\Delta u=f$ on a square with prescribed Dirichlet boundary data, using the same spacing $h$ in both directions.

For comparison, the five-point Laplacian has expansionIts leading error is not a multiple of ${\Delta }^{2}u$, because the mixed fourth derivative is missing. To obtain an isotropic leading error, consider a symmetric nine-point finite-difference stencil with weight $a$ on each axial neighbor, $b$ on each diagonal neighbor, and $c$ on the center, all divided by $h^2$. Vanishing on constants and consistency of a numerical method require $4a+4b+c=0$ and $a+2b=1$. Taylor expansion then gives the fourth-derivative contributionFor this to equal $h^2{\Delta }^{2}u/12$ we must choose $b=1/6$, hence $a=2/3$, $c=-10/3$. The resulting nine-point finite-difference stencil operator isIts expansion, retaining the next term for later use, isThus $D_9$ by itself is second order on a general function. On a Poisson equation solution, however, ${\Delta }^{2}u=\Delta f$. Approximating this known source derivative by $D_{5}f$ gives the fourth-order Mehrstellen method equationEquivalently, its two stencils areThe normalized local truncation error is $O(h^4)$, since $D_{5}f-\Delta f=O(h^2)$. If the equation is multiplied by $6h^2$, the row residual is $O(h^6)$; this change of normalization does not make the solution sixth order. The fourth-order correction of the nine-point Poisson stencil retains only the closest axial and diagonal unknowns. A fourth-order one-dimensional second derivative instead uses $(-U_{i+2}+16U_{i+1}-30U_i+16U_{i-1}-U_{i-2})/(12h^2)$, widening the unknown stencil and introducing negative outer weights. Mehrstellen method accuracy is obtained by correcting known source values rather than widening that unknown stencil.

The compact scheme has useful stability and solvability properties. With homogeneous Dirichlet boundary data, $-D_9$ is symmetric, has positive diagonal and nonpositive off-diagonal entries, and is a positive definite symmetric operator. Indeed its quadratic form is a sum of positive edge weights times squared differences, with boundary values set to zero. Vanishing of this sum would force a constant on the connected grid and then zero by connection to the boundary. Consequently the Dirichlet boundary condition linear system has a unique solution. The same weights give a discrete maximum principle: if $D_{9}v\ge 0$ at every interior node and $v\le 0$ on the boundary, a positive maximum inside would make every weighted difference to its neighbors nonpositive. Equality forces the maximum to propagate to the boundary, which is impossible. Thus $v\le 0$ everywhere.

This also turns consistency of a numerical method into a quantitative convergence result, rather than merely suggesting an order. Let $B_{h}f=f+h^2{D}_{5}f/12$, let $\tau =D_{9}u-B_{h}f$ be the normalized residual of the exact solution, and set $e=U-u$. Exact Dirichlet boundary data imply $e=0$ on the boundary and $D_{9}e=-\tau$. On the unit square the quadratic barrierhas $D_{9}q=-1$ exactly, is nonnegative on the boundary, and has maximum $1/8$. If $\Vert \tau {\Vert }_{\infty }\le \epsilon$, then $D_9(e-\epsilon q)\ge 0$ and $D_9(-e-\epsilon q)\ge 0$, while both comparison functions are nonpositive on the boundary. The discrete maximum principle provesThis is maximum-norm convergence of the corrected nine-point Poisson scheme. It requires sufficient smoothness of the solution up to the boundary for the uniform truncation bound. On other domains the source and boundary closures must retain the required accuracy; boundary singularities can invalidate a smooth-solution order estimate.

The same expansion reveals how to add still more accuracy. Differentiating $\Delta u=f$ givesThus the sixth derivatives appearing in the $h^4$ term can also be expressed solely through the source. With exact source derivatives,is the sixth-order source correction of the nine-point Poisson stencil, with normalized residual $O(h^6)$. If source derivatives are replaced by differences, $\Delta f$ must be approximated through fourth order and the fourth derivatives through second order to retain that accuracy. The unknowns still occupy the same nine-point finite-difference stencil; only evaluation of known source data becomes more elaborate. The same inverse bound yields sixth-order convergence when these approximations and the boundary data have matching accuracy.

A particularly instructive case is the Laplace equation, $f=0$. Both displayed error corrections automatically vanish: ${\Delta }^{2}u=0$ and $u_{xxxxyy}+u_{xxyyyy}=\Delta u_{xxyy}=0$, which also makes the pure sixth derivatives sum to zero. Expanding the stencil two orders further gives, for a sufficiently smooth harmonic function,For verification, the three eighth-derivative contributions are $(u_{xxxxxxxx}+u_{yyyyyyyy})/20160$, $(u_{xxxxxxyy}+u_{xxyyyyyy})/2160$ and $u_{xxxxyyyy}/864$. Because $u$ is a harmonic function, their sum is $u_{xxxxxxxx}(1/10080-1/1080+1/864)=u_{xxxxxxxx}/3024$. The example $u=\mathrm{Re}(x+iy{)}^8$ has $D_{9}u(0,0)=40h^6/3$, so this term is genuinely present. The unmodified harmonic nine-point scheme therefore has sixth-order nodal convergence under the same smoothness and boundary assumptions. This harmonic superconvergence of the nine-point stencil is special to harmonic functions; it should not be confused with the fourth-order source-corrected scheme for general Poisson equation data.

The appeal of Mehrstellenverfahren is this combination of a compact sparse matrix, favorable discrete maximum principle properties, and high accuracy obtained by using the governing equation. The derivation also shows precisely which accuracy belongs to the operator on arbitrary functions, which belongs to its action on solutions, and which survives in the computed boundary-value solution.

Interpret the square as $[0,1{]}^2$, use equal spacing $h=1/J$ in both directions and retain interior indices $1\le m,n\le J-1$. Boundary values contribute to the right-hand side and do not change the interior coefficient matrix. Products of the one-dimensional sine modes used in the discrete sine transformform a basis, and substitution into the five-point Laplacian givesThe interior matrix is ${\Delta }_h+\lambda I$, so it is nonsingular exactly whenThese are the discrete Helmholtz resonance values, including their possible multiplicities. A sufficient condition is $\lambda <{\Lambda }_{11}=8h^{-2}{\sin }^2(\pi /(2J))$, so in particular every nonpositive $\lambda$ is allowed. That is not a necessary restriction: positive values between, or above, the finitely many discrete eigenvalues are nonsingular as well. With homogeneous Dirichlet boundary conditions a nonresonant system has only the zero solution; at a resonance the corresponding modes of the discrete sine transform give nonzero homogeneous solutions.

Take the boundary-fitted grid $h=1/(M+1)$ and assume the standard consistency of a numerical method hypothesis $u\in C^4(\overline{\Omega })$ with bounded fourth partial derivatives. Taylor expansion along each coordinate, including a bounded fourth-order remainder, gives the five-point Laplacian truncation errorThe numerical equation and exact grid values therefore imply $A_h\mathbf{e}=\mathbf{\tau }$, where $A_h=-{\Delta }_h$ with zero boundary values. Its discrete sine transform eigenvectors are $\sin (\pi rih)\sin (\pi sjh)$, $1\le r,s\le M$, and its eigenvalues areThe one-dimensional sine vectors form an orthogonal basis, hence so do their tensor products. The least eigenvalue occurs at $r=s=1$. Since $\sin (\pi h/2)\ge h$ for $0<h\le 1$, ${\lambda }_{11}\ge 8$, so the Euclidean operator norm of $A_h^{-1}$ is at most $1/8$. There are $M^2$ residual components, and $Mh<1$, givingThe factor $M$ matters: pointwise second-order consistency of a numerical method becomes first order in the unweighted Euclidean grid norm.

The PDF does not explicitly impose regularity on the exact solution. Without such a hypothesis its unrestricted error claim is false. Here is a counterexample, with a classical solution smooth in the open square and continuous with zero boundary values:The sampled numerical solution separates as $v_i\sin (\pi jh)$. Its error $e_i=v_i-g(ih)$ solves $Ae=\tau$, where $A=-{\Delta }_{h,x}+{\lambda }_{y}I$ and ${\lambda }_y=4h^{-2}{\sin }^2(\pi h/2)$. The one-dimensional residual is ${\tau }_i={\Delta }_{h,x}g(ih)-g^{\prime \prime }(ih)+({\pi }^2-{\lambda }_y)g(ih)$. At the first node,The leading coefficient is strictly negative. Moreover $g^{\prime \prime \prime \prime }<0$ on $(0,1)$, so the triangular-average formula for the second difference and concavity of $g^{\prime \prime }$ give ${\Delta }_{h,x}g\le g^{\prime \prime }$. Thus the positive part of every residual is at most $C{h}^2$. The matrix $A$ has a nonnegative inverse, and its first diagonal inverse entry is at least $1/(2h^{-2}+{\lambda }_y)$, as follows from the nonnegative Neumann expansion of its tridiagonal matrix. The discrete comparison principle bounds the inverse image of the residual's positive part by $C{h}^2$, while the negative first residual gives $e_1\le -c\sqrt{h}+C{h}^2$. Hence $|e_1|\ge c^{\prime }\sqrt{h}$ for small $h$. Since ${\sum }_{j=1}^M{\sin }^2(\pi jh)=1/(2h)$, the full Euclidean error is bounded below by a positive constant. It cannot be $O(h)$. Thus the boxed estimate holds for the intended smooth solution class; a regularity condition is genuinely necessary.

The five-point Laplacian applied to $f$ has expansionThe corrected right side $h^{2}f+(h^2/12){\Gamma }_{5}f$ cancels the $h^4\Delta f/12$ term in ${\Gamma }_{9}u$. ConsequentlyMore precisely, the unscaled residual's $h^6$ coefficient is $-(u_{xxxxxx}+u_{yyyyyy})/240+(u_{xxxxyy}+u_{xxyyyy})/144$. This fourth-order correction of the nine-point Poisson stencil does not require $f$ to be harmonic.

The five-point Laplacian on the square grid isIt adds the second-order central differences in the two coordinate directions and uses the given zero Dirichlet boundary conditions when a neighbouring node lies on the boundary.

The sine addition formula impliesApply this identity with $\alpha =m\pi h$ in the $x$ direction and with $\alpha =n\pi h$ in the $y$ direction. For $U_{i,j}^{(m,n)}=\sin (m\pi ih)\sin (n\pi jh)$, it givesHence the discrete eigenvalues areBoth sine factors vanish on their respective boundary sides. Also $U_{1,1}^{(m,n)}>0$, since $m\pi /N$ and $n\pi /N$ lie strictly between zero and $\pi$. Thus each array is a nonzero discrete eigenvector, rather than merely the zero solution of the homogeneous system. The separated structure is also described by the five-point Dirichlet Laplacian as a Kronecker sum.