Neutron STEM Lab
Virtual engineering lab

Mesh Refinement & Solver Verification

Solve a manufactured 2D Poisson problem on three nested meshes and compare exact error, residuals, observed order, and GCI.

Fluid Mechanics / Numerical MethodsintermediateValidated educational model
Learning mode

Guided laboratory

Use the checklist when you want a structured lab. Explore mode leaves the instruments unrestricted.

Set up → solve → verify

Build your numerical experiment

Read the equations

Start with an example. Change one setting at a time and compare accuracy, conservation, and computation cost.

Boundary conditions. Zero values on all four edges of a unit square; smooth manufactured forcing and known exact solution.

unitless
Even count; meshes use N, 2N, and 4N cells.
unitless
A loose iteration stop can hide spatial convergence.
unitless
Reaching the cap is reported explicitly.

Choose an example or use the defaults, then run. Larger grids may take several seconds.

Keep this experiment

Valid settings are saved on this device. Project JSON preserves physical inputs and display units. Restore it and run again for fresh results.

Settings stay in this browser. Simulations run locally.

Measurements

Instrument readings

Fine-grid RMS error
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Recorded from the current completed simulation.
Fine-grid relative residual
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Recorded from the current completed simulation.
Theory

Equations and model

A known manufactured solution

Choose a smooth exact solution and derive the source. Central five-point differences are second-order accurate. SOR and Gauss–Seidel solve the same discrete equations with different convergence speeds.

−∇2ϕ=f,ϕ=sin⁡(mπx)sin⁡(πy),m∈{1,3}-\nabla^2\phi=f,\quad\phi=\sin(m\pi x)\sin(\pi y),\quad m\in\{1,3\}f=π2(m2+1)sin⁡(mπx)sin⁡(πy)f=\pi^2(m^2+1)\sin(m\pi x)\sin(\pi y)ERMS=1N∑k=1N(ϕk−ϕexact,k)2E_{\mathrm{RMS}}=\sqrt{\frac1N\sum_{k=1}^N(\phi_k-\phi_{\mathrm{exact},k})^2}

Separate grid and iteration error

Three meshes use equal refinement ratio r = 2. Center values define the observed order, Richardson extrapolation, and fine-grid convergence index. Estimates are withheld for loose iterative convergence or non-monotone differences. GCI does not cover all physical uncertainty.

p=ln⁡∣Qc−QmQm−Qf∣ln⁡r,Q∞=Qf+Qf−Qmrp−1p=\frac{\ln\left|\frac{Q_c-Q_m}{Q_m-Q_f}\right|}{\ln r},\qquad Q_\infty=Q_f+\frac{Q_f-Q_m}{r^p-1}GCIf=1.25 ∣(Qf−Qm)/Qf∣rp−1×100%\mathrm{GCI}_f=\frac{1.25\,|(Q_f-Q_m)/Q_f|}{r^p-1}\times100\%
Validation

Independent checks

Numerical balance or equation residualNot run

Review conservation, boundary, entropy, or residual checks.

Expected
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Simulated
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Error
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Tolerance
0%

Run the experiment to perform this check.

Reference, bounds, or convergenceNot run

Review the specific check and its limits; this is not a universal accuracy certificate.

Expected
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Simulated
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Error
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Tolerance
0%

Run the experiment to perform this check.

Engineering interpretation

Run the experiment to generate an engineering interpretation.

Assumptions and limitations
Assumptions
  • Zero values on all four edges of a unit square; smooth manufactured forcing and known exact solution.
  • Constant properties; uniform one- or two-dimensional meshes.
  • Physical quantities stored in SI. Changing display units preserves the simulation.
Limitations
  • Educational numerical verification does not certify a physical design.
  • No arbitrary CAD geometry, turbulence, compressibility, multiphase flow, or free surfaces.
  • Grid resolution, bounded computation, and reference limits are reported explicitly.
Local experiment export

Save your measurements and setup

Exports are generated in your browser. No account or server upload is required.

Settings JSON
Current parameters and instrument controls, including waveform, output enable, scope coupling, timebase, trigger, and cursors when present. Data labs also include the dataset, mappings, exclusions, budget, or propagation setup.
Measurements CSV
Completed instrument readings with labels, units, and descriptions.
Plot PNG
The current canvas-based scope, transient, sweep, or statistics visualization when available.
Summary
Markdown report with objectives, setup, measurements, validation, interpretation, assumptions, and graded guided concept-check results.

Ready to export the current local experiment state.

Learning objectives

What this experiment should establish

  • Verify a known solution.
  • Measure observed order.
  • Test iteration limits.
  • Resolve a shorter wave.
Check your understanding

Questions to answer from the experiment

  1. What does a small equation residual show?
  2. For second-order asymptotic accuracy, halving spacing reduces error by about what factor?
  3. Why withhold extrapolation for poorly converged solves?
  4. What does a manufactured solution mainly verify?
Continue learning
CFD LaboratoryExplore the other numerical experiments.Fluid mechanics toolsConnect numerical results to fluid mechanics calculations.Data Analysis & Uncertainty LabAnalyze exported numerical data.

Sources and model review

Reviewed 2026-09-19. The educational model exposes its assumptions and validation; source references do not imply external certification.