Electrostatic Poisson solvers and boundaries

For one-dimensional SI electrostatics, minus the second derivative of potential equals charge density divided by permittivity. A centered difference gives a positive diagonal and negative neighbor coefficients when written as a positive definite Dirichlet matrix. Boundary potential terms move to the right-hand side for interior unknowns. Multiplying the equation by the squared spacing makes every matrix-equation component have potential units; retaining the divided form instead gives a residual with potential per length squared units.

The electric field is minus the potential gradient. It may be stored at cell faces or computed by a specified centered node difference; these choices have different boundary and Gauss-law interpretations. Our hand calculation uses centered node fields only as a declared diagnostic. A physical electrode potential, an imposed normal flux and a periodic identification constrain different quantities. Choosing a zero potential on a boundary is not generally equivalent to choosing the mean potential as a gauge in a periodic domain.

Write the discrete equations as Aφ = b and an approximate solution as φtilde. Its residual is b minus Aφtilde, while its algebraic solution error is the inverse matrix applied to that residual. A small residual is useful only relative to consistent norms, units and the conditioning of A. Refining the mesh can worsen the matrix condition number, so the same unscaled residual threshold need not imply the same potential error on every grid.

Jacobi iteration updates each unknown from the previous iteration's neighbors. It is simple to inspect but often slow for long-wavelength error, motivating multigrid or other preconditioned methods on larger problems. Iteration count is not physical elapsed time. Once algebraic error is small compared with truncation error, further iterations do little to improve the continuum approximation. A separate grid study, with consistent boundary data, measures the error remaining from replacing derivatives and geometry by discrete representations.

Summing a periodic discrete Laplacian cancels all neighbor differences. Its source must therefore have zero mean. A pure Neumann continuum problem has a corresponding relation between integrated source and outward normal derivative; if it holds, an additive constant remains arbitrary. Fixing one value or the mean removes that nullspace but cannot make an incompatible source solvable. Silently subtracting the source mean changes the physical problem unless a neutralizing background was intentionally part of the model.

Residuals and electric fields can be insensitive to a constant potential shift, which is why the gauge should be stated in reproducibility records. Boundary discretization also affects force accuracy near electrodes even when an iterative solve is exact. The optimization lab illustrates a convergent or divergent scalar iteration with a selectable step parameter. It has no Poisson matrix, spatial mesh, electrodes, nullspace or charge source and therefore cannot verify this electrostatic boundary-value solver.