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The diagrams solve an explicitly bounded Dirichlet problem in a uniform charge-free medium of constant permittivity. Take a width scale $W=1$ and terminal electric potential difference $V=1$. Two zero-thickness ideal plate segments occupy $|x|\le W/2$ at $y=\pm d/2$, with electric potentials $\pm V/2$. Fix $\varphi =0$ on the entire rectangle $x=\pm 2W$, $y=\pm 1.5W$. This boundary is an additional enclosing equipotential, rather than a condition at infinity. The model is invariant along the omitted third direction. The connection between the electric potential, its gradient, and the charge-free Laplace equation follows from electrostatics.^[ref]

On a square mesh of spacing $h=W/160$, the finite difference method imposesat free nodes, while plate and outer nodes retain their prescribed electric potentials. There are $641\times 481$ nodes. Each plotted gap aligns exactly with the mesh; the smallest gap has 16 mesh intervals. The generators solve the resulting linear system by the Conjugate gradient method, stopping when the maximum dimensionless five-point residual is below $2\times 10^{-10}$, and use central finite differences to approximate $\mathbf{E}=-\nabla \varphi$ away from plate nodes.

The discrete maximum principle keeps the computed electric potential between $-V/2$ and $+V/2$. Reflection checks require evenness in $x$ and oddness in $y$; halving $h$ tests discretization effects, while enlarging the outer rectangle tests sensitivity to its artificial boundary conditions. These are different checks: a small residual only establishes that the discrete equations were solved. Differentiating near zero-thickness endpoints is especially sensitive to the mesh, so the plots do not report a converged edge peak. No total electric charge, exact open-domain three-dimensional capacitance, or measured material property is extracted from this model.

For a bounded numerical check, compare $h=W/80$ with the plotted $h=W/160$ at midplane positions $x/W=0,0.25,0.40,0.50,0.60,0.80$. At $d/W=0.10$, the largest change in $-E_{y}d/V$ is approximately $0.0265$, at $x/W=0.50$; the other five sampled changes are below $0.00472$. At $d/W=0.25$, moving the outer rectangle from $\pm 2W\times \pm 1.5W$ to $\pm 3W\times \pm 2W$ while retaining $h=W/80$ changes these samples by at most approximately $0.00622$. These calculated sensitivities support the qualitative comparison and expose its limits; they are not experimental uncertainties or evidence that the artificial boundary has converged to infinity.

For the square-grid nine-point finite-difference stencil for the Laplacian,If $\Delta u=0$, both displayed correction terms vanish: the mixed sixth derivatives sum to $\Delta u_{xxyy}=0$, and the pure sixth derivatives sum to the negative of that sum. Expanding two orders further yields $D_{9}u=h^6{u}_{xxxxxxxx}/3024+O(h^8)$ for a smooth harmonic function. Thus the compact harmonic scheme has sixth-order normalized consistency of a numerical method and, with the discrete maximum principle inverse bound, sixth-order nodal convergence on a square with exact smooth Dirichlet boundary data. For example $\mathrm{Re}(x+iy{)}^8$ shows that the sixth-order term need not vanish.

On the unit square with exact Dirichlet boundary data, let $D_{9}U=B_{h}f$ and suppose the normalized residual on the exact smooth solution has supremum norm at most $\epsilon$. The error satisfies $D_{9}e=-\tau$ and vanishes on the boundary. The barrier $q=[x(1-x)+y(1-y)]/4$ has $D_{9}q=-1$, $q\ge 0$ on the boundary, and maximum $1/8$. Apply the discrete maximum principle to $e-\epsilon q$ and $-e-\epsilon q$, obtaining $|e|\le \epsilon q$. Thus $\Vert e{\Vert }_{\infty }\le \epsilon /8$, and fourth-order residual gives fourth-order supremum norm nodal convergence. The proof needs a uniform smoothness bound and a compatible boundary discretization.

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.

Let $U_m^n=u(mh,nk)$ be the sampled exact solution and let $A_h$ denote the update matrix of the Forward Euler diffusion scheme. For $0\le \mu \le 1/2$,is a convex combination, with the homogeneous boundary values included. ThereforeThis is the parabolic discrete maximum principle and proves max-norm stability directly.

The Taylor theorem and the second-order central difference give the exact-grid residualfor a sufficiently smooth solution on a fixed interval $0\le t\le T$. If $e^n=U^n-u^n$, thenIteration for $nk\le T$ yieldsThus $\mu \le 1/2$ implies convergence when the initial grid values converge to the initial data.

Let $h=\Delta x$. A one-dimensional Taylor expansion givesAdding the three coordinate directions shows that the exact solution has stencil defectThis is the local defect of the seven-point Dirichlet Laplacian. Its inverse has max-norm size $O(h^{-2})$: this follows from the discrete maximum principle, for example by comparison with the grid restriction of $x_1(1-x_1)+x_2(1-x_2)+x_3(1-x_3)$, whose stencil is $-6h^2$. Therefore the grid error is $O(h^2)$.

The question writes this error as $O(h^{p+1})$, soIn the more usual terminology, the finite difference method is second-order accurate.

The nine-point finite-difference stencil for the Laplacian has expansion $D_{9}u=\Delta u+h^2{\Delta }^{2}u/12+h^4(u_{xxxxxx}+u_{yyyyyy}+5u_{xxxxyy}+5u_{xxyyyy})/360+O(h^6)$. If $\Delta u=f$, differentiating this equation gives $u_{xxxxyy}+u_{xxyyyy}=f_{xxyy}$ and $u_{xxxxxx}+u_{yyyyyy}={\Delta }^{2}f-3f_{xxyy}$. Hence the corrected compact equationhas sixth-order normalized consistency of a numerical method. Source derivatives can be supplied analytically; numerical approximation of $\Delta f$ must be fourth order and of the fourth derivatives second order. With exact square-grid Dirichlet boundary data and uniformly smooth solution, the discrete maximum principle inverse bound converts this residual into sixth-order nodal convergence. The unknown stencil stays compact even if more distant known source values are used.