Numerical boundary problems

Electrostatic boundary-value problems determine a potential inside a region from charge density, material properties, and conditions imposed on the boundary. When geometry is too complicated for a closed-form solution, the region can be discretized and the potential values solved numerically.

A solver that stops changing has not necessarily found the continuum solution. It may have converged only to its discrete equations, used incomplete boundary conditions, or stopped because floating-point updates became small. Iteration error and grid error need separate evidence.

Poisson and Laplace equations

In vacuum electrostatics,

\[\nabla^2V=-\frac{\rho}{\epsilon_0}\]

Here \(V\) is in volts, \(\rho\) is in coulombs per cubic meter, and \(\epsilon_0\) is the vacuum permittivity.

In a charge-free region,

\[\nabla^2V=0\]

Zero charge density inside the region does not imply zero field. Charges outside and conductors on the boundary determine the boundary values and can produce a nonconstant potential throughout the charge-free interior.

For a spatially varying linear dielectric, the more general equation is

\[\boldsymbol{\nabla}\cdot \left(\epsilon\boldsymbol{\nabla}V\right) =-\rho_{\mathrm{free}}\]

Pulling permittivity outside the derivative is valid only when it is constant in the region.

Boundary conditions select the solution

A Dirichlet boundary condition specifies potential:

\[V|_{\partial\Omega}=V_b\]

A conductor held at a known voltage is a common example. A Neumann condition specifies the normal derivative, which is related to the normal electric field:

\[\frac{\partial V}{\partial n} =-\mathbf E\cdot\hat{\mathbf n}\]

A mixed problem uses different condition types on different boundary pieces.

Dirichlet conditions generally fix the electrostatic potential uniquely for a well-posed linear problem. Pure Neumann conditions determine potential only up to an additive constant and require a compatibility condition connecting source charge with prescribed boundary flux. One reference potential must then be chosen.

Missing or contradictory constraints cannot be repaired by more iterations. Boundary type, outward-normal convention, and units are part of the physical problem.

One-dimensional analytic fixture

Consider a source-free interval from \(x=0\) to \(x=L\) with fixed endpoint potentials:

\[\frac{d^2V}{dx^2}=0, \qquad V(0)=V_L, \qquad V(L)=V_R\]

The exact solution is

\[V(x)=V_L +\frac{V_R-V_L}{L}x\]

If the endpoints are zero volts and ten volts, the midpoint is five volts. This fixture tests boundary indexing, interpolation, and sign without requiring an approximate reference.

Finite-difference Laplacian

On a uniform square grid with spacing \(h\), the five-point approximation is

\[\nabla^2V_{i,j} \approx \frac{ V_{i+1,j}+V_{i-1,j} +V_{i,j+1}+V_{i,j-1} -4V_{i,j}} {h^2}\]

For Laplace's equation, solving for the center value gives

\[V_{i,j} =\frac14\left( V_{i+1,j}+V_{i-1,j} +V_{i,j+1}+V_{i,j-1} \right)\]

Each interior potential equals the average of its four nearest neighbors. This averaging property is a consequence of Laplace's equation on this grid, not a universal smoothing rule for every partial differential equation.

For Poisson's equation with constant permittivity,

\[V_{i,j} =\frac14\left( V_{i+1,j}+V_{i-1,j} +V_{i,j+1}+V_{i,j-1} +\frac{h^2\rho_{i,j}}{\epsilon} \right)\]

The positive source term in the update follows from the negative sign in Poisson's equation. A manufactured solution with known charge density can catch a sign error.

Jacobi and Gauss–Seidel relaxation

Jacobi iteration computes every new interior value from the previous full grid, then swaps grids. It is easy to reason about and parallelize. Gauss–Seidel iteration updates in place and immediately uses newly updated neighbors, usually converging faster but making update order visible.

Successive over-relaxation can accelerate convergence further by moving partway beyond the Gauss–Seidel update. An unsuitable relaxation parameter can slow convergence or make the iteration diverge.

The method and update order affect the path to the discrete solution. For a converged well-posed linear problem, they should not change that solution beyond tolerance.

Residual measures equation satisfaction

For constant permittivity, define the discrete Poisson residual

\[R_{i,j} =(\nabla_h^2V)_{i,j} +\frac{\rho_{i,j}}{\epsilon}\]

A common stopping measure is the maximum absolute interior residual:

\[\lVert R\rVert_\infty =\max_{i,j}|R_{i,j}|\]

The residual has units of volts per square meter. A tolerance therefore needs units or a dimensionless normalization based on a declared voltage and length scale.

A small update between iterations is not the same as a small residual. Slowly converging methods can make tiny updates while significant low-frequency error remains. Calculate the equation residual directly.

Iteration convergence versus mesh convergence

Driving the residual below tolerance solves the discrete equations on one grid. It does not measure the difference from the continuum solution. Grid refinement addresses that second question.

Solve the same physical geometry on spacings \(h\) and \(h/2\), using comparable residual tolerances. Compare potential and derived field at the same physical coordinates.

For a smooth solution and a second-order stencil, error often decreases approximately by a factor of four when spacing is halved. Curved stair-step boundaries, source discontinuities, singular corners, or lower-order boundary stencils can reduce that rate.

Curved boundaries and material interfaces

A Cartesian grid approximates a curved conductor by stair steps unless a cut-cell, finite-element, or boundary-fitted method is used. Refining the grid can improve the geometry, but the boundary representation must be part of the convergence study.

At an interface where permittivity changes, discretize the flux

\(\epsilon\boldsymbol{\nabla}V\)

consistently across cell faces. Simply assigning one cell-center permittivity to the ordinary constant-coefficient stencil can violate the normal displacement-field condition.

Surface charge can create a jump in normal field while potential remains continuous. Averaging potentials across the interface without respecting flux can smear the physical condition.

Deriving the electric field

After solving potential, calculate

\[\mathbf E=-\boldsymbol{\nabla}V\]

Differentiation amplifies grid noise, and field accuracy can be lower than potential accuracy. Use centered differences in smooth interiors and documented one-sided or interface-aware formulas near boundaries.

Conductor surfaces often have the strongest gradients. A visually smooth potential can still produce a poor surface field if the grid does not resolve curvature.

A boundary-preserving relaxation procedure

The algorithm must never treat constrained cells as ordinary unknowns.

import numpy as np

# Laplace equation in a square: top boundary 1 V, other edges 0 V.
voltage = np.zeros((21, 21))
voltage[-1, 1:-1] = 1.0
for iteration in range(20000):
    previous = voltage.copy()
    voltage[1:-1, 1:-1] = 0.25 * (
        previous[2:, 1:-1]
        + previous[:-2, 1:-1]
        + previous[1:-1, 2:]
        + previous[1:-1, :-2]
    )
    residual = np.max(
        abs(
            4 * voltage[1:-1, 1:-1]
            - voltage[2:, 1:-1]
            - voltage[:-2, 1:-1]
            - voltage[1:-1, 2:]
            - voltage[1:-1, :-2]
        )
    )
    if residual < 1e-9:
        break
else:
    raise RuntimeError("Relaxation reached the iteration limit.")
assert np.all(voltage[-1, 1:-1] == 1.0)
print("Iterations:", iteration + 1, "centre voltage:", voltage[10, 10])

This version describes Jacobi iteration because each update reads the copied grid. Gauss–Seidel should document that it reads the partially updated grid instead.

Validation cases

Begin with constant boundary values and no charge; the solution should be constant. Then use the one-dimensional linear fixture embedded in the grid. A manufactured quadratic potential can provide a known Poisson source and test its sign.

Check that every constrained node remains exact after every iteration. Start from different interior guesses and confirm convergence to the same solution. Record residual history and fail explicitly when the iteration cap is too small.

Finally refine the mesh at fixed physical probes. Compare potential, field components, boundary flux, and total enclosed charge from Gauss's law. A converged residual, correct boundaries, and stable refinement together provide much stronger evidence than a plot that merely stops changing.

A worked Python notebook

Use the one-dimensional Laplace solution between boundaries at 0 and 10 volts separated by one metre. The question for this calculation is precise: how does potential between plates change when we change horizontal position, while keeping the other choices fixed? The earlier sections explain the physical relationship. Here we turn one part of that relationship into a calculation that you can run, inspect, and change in Google Colab.

The region contains no charge and has uniform permittivity. Keep that sentence beside the results. A formula can return a number outside its useful range, and a plotting library will usually draw that number without asking whether the physical assumptions still hold. The selected interval is from 0 to 1 m. It gives us a specific domain to discuss instead of leaving the graph's range to a software default.

Use the notebook link above the lesson, run the setup cell, and then run the cells in order. The code below is also included there. NumPy handles the arrays of numerical values, SymPy keeps the equation symbolic, and Matplotlib draws the results. If these libraries or notebook cells are unfamiliar, start with Your first physics notebook.

State the relationship before choosing a graph

In this small example, q stands for horizontal position, expressed in m. The output is potential between plates, expressed in V. The equation below uses the numerical values in those units. Any fixed coefficients therefore carry the units needed to make the relation consistent. A value supplied in another unit must be converted before it enters this expression.

\[f(q) = 10 q\]

This is a function of one changing input. It is not a statement that the physical system has only one variable. We deliberately hold the other inputs fixed so that one cause of variation is visible. Changing several inputs at once would make it harder to tell which change produced the output difference. Write down the held-fixed choices before trying a different experiment.

At 0.5 m, the expression gives 5 V. This is our reference calculation. It is a prediction of the stated example, not a new measurement. Its displayed digits help us compare implementations. They should not be read as a claim that physical inputs are known to six significant figures.

import numpy as np
import sympy as sp
import matplotlib.pyplot as plt

q = sp.Symbol("q", real=True)
expression = sp.sympify("10*q", locals={"q": q})
reference_input = 0.5
reference_output = float(expression.subs(q, reference_input))
print("Input:", reference_input, "m")
print("Predicted potential between plates:", reference_output, "V")

sp.Symbol declares a mathematical symbol. It does not assign a numerical measurement to that symbol. subs then replaces the symbol with our chosen input, and float converts the resulting symbolic number into a standard Python numerical value. Keeping these steps separate lets us inspect the equation before numerical substitution. It also prevents a later plot from becoming the only surviving record of the model.

Compare a few cases you can explain

Before drawing hundreds of points, calculate three. The lower endpoint gives 0 V. The midpoint, at 0.5 m, gives 5 V. The upper endpoint gives 10 V. Read these together with the assumptions: they describe different choices of horizontal position within the same example, with the other inputs held fixed.

These three values are a useful first check, but they do not establish the shape between them. Two endpoints can be equal even when the curve bends or oscillates in between. Likewise, a large change between endpoints does not tell us whether the relationship is linear. We need the equation and intermediate samples to answer those questions.

evaluate = sp.lambdify(q, expression, modules="numpy")
inputs = np.array([0.0, 0.5, 1.0])
outputs = np.asarray(evaluate(inputs), dtype=float)
outputs = np.broadcast_to(outputs, inputs.shape)
for input_value, output_value in zip(inputs, outputs):
    print(f"{input_value:.6g} m -> {output_value:.6g} V")
assert np.all(np.isfinite(outputs))

lambdify turns the symbolic expression into a numerical function. We explicitly select NumPy as its numerical backend so it can accept an array. The same expression now has two useful forms: one for algebra and one for calculating many cases. The assertion checks that the chosen inputs produce finite outputs. It would catch an infinity or an undefined numerical result, but a finite value can still be physically inappropriate.

Make a graph that answers the question

The horizontal axis shows horizontal position; the vertical axis shows potential between plates. Every plotted point is a prediction of the same expression. We mark the reference case separately so that a reader can connect the earlier printed value to its place on the curve. The axis labels include units because a shape without a scale cannot tell us the size of a physical effect.

input_grid = np.linspace(0.0, 1.0, 301)
output_grid = np.asarray(evaluate(input_grid), dtype=float)
output_grid = np.broadcast_to(output_grid, input_grid.shape)
fig, ax = plt.subplots(figsize=(7, 4))
ax.plot(input_grid, output_grid, label="Stated model")
ax.scatter([reference_input], [reference_output], label="Reference case", zorder=3)
ax.set_xlabel("Horizontal position (m)")
ax.set_ylabel("Potential between plates (V)")
ax.legend()
ax.grid(alpha=0.25)
fig.tight_layout()
plt.show()

Look for a sign change, a turning point, or a region where the curve becomes steep. Connect each feature to the expression rather than to the colour of the line. If the curve has a minimum or maximum, changing the input in opposite directions may have similar effects on the output near that point. If it is nearly straight over a short interval, a local linear approximation may be useful even when the complete expression is nonlinear.

Increasing the number of samples makes the line smoother and may reveal features that a coarse grid missed. It does not improve the physical inputs or enlarge the model's validity range. For quantities that can take only discrete values, use the line as a guide to the relationship and interpret only the allowed input values. A plotting function does not enforce such restrictions for you.

Measure the local sensitivity

A derivative asks a more specific question than the whole graph: how quickly does potential between plates change near the reference input? Its units are the output unit divided by the input unit. For this calculation, the derivative at 0.5 m is 10 in those ratio units. Locally, the output increases as the input increases. This statement is local; the sign or size of the slope can be different elsewhere on the curve.

\[f'(q) = 10\]

For a small input change, multiply the derivative by that change to estimate the output change. Then compare that estimate with a fresh evaluation of the full expression. The distinction between these two calculations matters: the derivative gives a tangent approximation, while substitution evaluates the chosen expression at the new point.

slope_expression = sp.diff(expression, q)
slope_at_reference = float(slope_expression.subs(q, reference_input))
changed_input = 0.505
linear_prediction = reference_output + slope_at_reference * (
    changed_input - reference_input
)
full_prediction = float(evaluate(changed_input))
print("Symbolic derivative:", slope_expression)
print("Tangent prediction:", linear_prediction, "V")
print("Full expression:", full_prediction, "V")
print("Approximation difference:", full_prediction - linear_prediction, "V")

Here the changed input is 0.505 m. The tangent estimate is 5.05 V, while the full expression gives 5.05 V. If the two values are close, it means the chosen input change is small enough for this local approximation at this point. It does not mean the entire model is linear. If the slope is zero, a quadratic or higher-order term may supply the leading change instead.

This is also the beginning of uncertainty propagation. If the input has a small stated standard uncertainty, the magnitude of the slope times that uncertainty estimates one contribution to output uncertainty. Do not automatically use the same rule for a wide interval, a discontinuity, a turning point with zero first derivative, or several correlated uncertain inputs. Those situations need more of the model than one slope.

Check the derivative with numerical differences

We can estimate the same slope from values on either side of the reference point. Subtract the lower output from the upper output and divide by twice the input step. Parentheses are essential: the entire output difference must be divided by the entire input interval. Without them, Python can execute a different mathematical expression while still producing a plausible number.

The calculation below uses three decreasing step sizes. Each estimate is compared with the symbolic derivative, so the reference is independent of the numerical difference formula. Near an endpoint, it is acceptable here to evaluate a small extension only because the displayed mathematical expression is smooth there; such a check does not assert that extrapolated values describe additional physical cases.

input_span = 1.0
for fraction in [1e-2, 1e-3, 1e-4]:
    step = input_span * fraction
    upper = float(evaluate(reference_input + step))
    lower = float(evaluate(reference_input - step))
    numerical_slope = (upper - lower) / (2 * step)
    difference = numerical_slope - slope_at_reference
    print(f"step={step:.6g}: slope={numerical_slope:.6g}, difference={difference:.6g}")

For a smooth function, reducing the step often reduces the approximation error at first. Very small steps can eventually make subtraction of nearly equal floating-point values less reliable. Therefore “smaller” is not an unlimited guarantee of “better.” Compare the observed behaviour over a useful range of steps instead of choosing the smallest number the computer can represent.

An absolute slope difference is useful when the expected slope is near zero. A relative difference would divide by a very small reference and could look enormous even when the absolute discrepancy is tiny. Away from zero, a relative difference can help compare quantities with very different scales. The tolerance should reflect the question and the numerical method, rather than a universal decimal threshold copied into every test.

Change the example and preserve the reasoning

Make one copy of the reference calculation before editing it. Choose a second value of horizontal position inside the stated interval, predict whether potential between plates will rise or fall, and then evaluate the expression. Place the prediction and the result in a text cell. If they disagree, identify whether the mistake was in the sign, the unit conversion, the algebra, or the assumption about how the curve behaves.

Next choose a shorter plotting interval around the reference point. Compare the local curve with its tangent approximation. Then widen the interval again. The difference shows why a useful local estimate need not describe the whole range. Keep the axis labels and numerical scale visible when comparing the plots; automatic axis limits can make two different effects appear equally large.

Finally, return to the physical limit stated at the start: The region contains no charge and has uniform permittivity. Describe one change to the setup that would violate that limit. State what new input, equation, or observation would be needed before trusting an answer in the changed situation. This last step connects the Python experiment to the physics instead of treating successful execution as the only criterion for a useful result.

Before saving, restart the runtime and run all cells in order. A result that depends on an earlier, now-deleted input cell is not reproducible from the saved notebook. Keep unrounded values for later calculations, round only the reported output, and label any new data as measured, assumed, or generated. The graph and printed values should remain traceable to the same expression and the same choices of inputs.

Recover the output change by integrating the slope

There is another way to check that the expression and its derivative describe the same relationship. Add up the small output changes predicted by the slope across the chosen input interval. In the continuous limit, this integral equals the output at the upper endpoint minus the output at the lower endpoint. The result has units of V: the input unit in the integration cancels the input unit in the denominator of the slope.

This statement is about the mathematics of this particular relationship. It does not assign a new physical meaning to an arbitrary area under a graph. For example, an area under force versus distance represents work, while an area under a different pair of quantities may represent something else or have no standard name. Always carry the axis units through the calculation before naming an integral.

slope_function = sp.lambdify(q, slope_expression, modules="numpy")
endpoint_change = float(evaluate(1.0) - evaluate(0.0))
for points in [101, 201, 401]:
    grid = np.linspace(0.0, 1.0, points)
    slopes = np.asarray(slope_function(grid), dtype=float)
    slopes = np.broadcast_to(slopes, grid.shape)
    integrated_change = np.trapezoid(slopes, grid)
    print(
        points,
        "integrated change:",
        integrated_change,
        "endpoint change:",
        endpoint_change,
    )

The trapezoidal rule joins neighbouring slope samples with straight segments and adds their signed areas. A negative contribution remains negative; replacing every slope by its magnitude would calculate a different quantity. Increasing the sample count provides a practical refinement check. If the values do not settle toward the endpoint difference, first inspect the interval, singularities, and array ordering.

Do not conclude that two agreeing calculations prove the model is physically complete. Both use the same selected relationship, so an omitted effect would be omitted from both. Their agreement is evidence that the numerical tools implement that relationship consistently. Comparing with an independent experiment or a more appropriate physical model is a separate test.

Keep this comparison in the notebook when you change the expression. It is particularly useful after adding a coefficient, changing a sign convention, or converting an input unit. A code cell that produces a smooth graph can conceal these mistakes; a second calculation of the same output change gives you another place to notice them.

Sources

• OpenStax, University Physics Volume 2, Section 7.4, “Determining Field from Potential”.
• OpenStax, University Physics Volume 2, Section 7.5, “Equipotential Surfaces and Conductors”.
• OpenStax, University Physics Volume 2, Section 6.4, “Conductors in Electrostatic Equilibrium”.
• Benjamin Crowell, Light and Matter, Chapter 22, “Fields”.
• Physics LibreTexts, “Electricity and Magnetism”.