Advanced Poisson Equation Inputs
Enter domain size, boundary values, source behavior, iteration settings, and a sample point.
The result appears above this form after submission.
Formula Used
The calculator solves a rectangular Dirichlet boundary problem using:
∇²u = f(x,y).
Depending on the selected sign, the right side is either
source / coefficient or -source / coefficient.
The finite difference form for an interior node is:
u(i,j) = [dy²(u(i+1,j)+u(i-1,j)) + dx²(u(i,j+1)+u(i,j-1)) - f dx²dy²] / [2(dx²+dy²)]
The solver then applies relaxation:
u_new = u_old + omega × (u_formula - u_old).
Iteration stops when the largest update is below tolerance or the iteration limit is reached.
How to Use This Calculator
- Enter the rectangular domain size in the X and Y directions.
- Select grid nodes. More nodes give better detail but slower solving.
- Set the four fixed boundary values.
- Choose a source model, amplitude, coefficient, and equation sign.
- Enter solver settings such as tolerance and relaxation factor.
- Choose a sample point for field and gradient reporting.
- Press calculate. Review the result section above the form.
- Download the CSV or PDF for reports and records.
Example Data Table
| Example |
Domain |
Grid |
Boundary values |
Source setting |
Expected behavior |
| Uniform loading |
1 × 1 |
31 × 31 |
All zero |
Constant, amplitude 10, sign - |
Positive interior hill with zero edges |
| Gaussian center |
1 × 1 |
41 × 41 |
All zero |
Gaussian, center 0.5,0.5 |
Smooth local peak near center |
| Side bias |
2 × 1 |
51 × 31 |
Left 5, right 0, top 0, bottom 0 |
Constant, amplitude 0 |
Field decays from left boundary |
| Sine benchmark |
1 × 1 |
41 × 41 |
All zero |
Sine, modes 1 and 1 |
Symmetric wave-like distribution |
Understanding Poisson Equation Analysis
What It Solves
Poisson equation appears when a field is shaped by a source term and boundary limits. It is common in electrostatics, heat transfer, gravity, fluid pressure, and image processing. The equation connects local curvature with forcing. When the source is positive, the surface bends one way. When it is negative, it bends the other way.
Numerical Method
This calculator uses a finite difference grid. The rectangular area is split into many small cells. Each interior value is updated from nearby values and the selected source. Boundary values stay fixed. The method repeats until the largest change is below the chosen tolerance or until the iteration limit is reached.
Grid and Relaxation
Grid density controls detail and speed. A larger grid gives a finer field map. It also needs more iterations and more memory. A smaller grid is useful for quick checks. Relaxation changes the update strength. Values near one are stable. Higher values can converge faster, but they may oscillate on difficult cases.
Source Models
The source model defines the right side of the equation. Constant loading is useful for uniform charge, heat generation, or body force studies. Gaussian loading creates a smooth local peak. Point loading approximates a concentrated source at one location. Sine loading helps test waves and benchmark examples.
Reading Results
Results include the computed field value at a chosen sample point. They also include estimated gradient components and residual error. The residual compares the final grid with the equation. A small residual suggests the numerical field is internally consistent. The heatmap helps reveal peaks, valleys, and boundary influence.
Reporting and Accuracy
Use the CSV export for spreadsheets, reports, or later checking. Use the PDF export for quick project notes. Always compare the answer with expected units and physical limits. Numerical results depend on grid size, tolerance, boundary data, and source scaling. Increase grid size slowly when accuracy is important. For teaching work, begin with simple square domains and equal boundaries. Then change one input at a time. This makes trends easier to see. For engineering work, record the assumptions with every export. The model is a numerical aid, not a replacement for validation, mesh studies, or expert review on safety critical designs. Use measured data when it is available.
FAQs
What is Poisson equation?
Poisson equation links the Laplacian of a field to a source term. It models potentials, temperatures, pressures, and other fields affected by internal forcing.
What boundary type is used here?
This calculator uses fixed value boundaries, also called Dirichlet boundaries. The four edges remain constant while interior values are solved iteratively.
What does the source coefficient mean?
The coefficient scales the source term. In electrostatic work, it can represent permittivity. In other models, it may represent material or field scaling.
Why does a larger grid take longer?
A larger grid has more unknown interior nodes. Each iteration updates more values, so solving needs more computation and sometimes more iterations.
What is the relaxation factor?
The relaxation factor controls update strength. A value of one gives standard Gauss-Seidel behavior. Higher values may speed convergence but can become unstable.
What is residual error?
Residual error checks how closely the final grid satisfies the finite difference equation. Smaller residuals usually indicate a more consistent numerical solution.
Can this solve real engineering problems?
It can support study and early estimation. For safety critical engineering, confirm assumptions, use verified tools, compare meshes, and seek expert review.
Why use CSV and PDF exports?
CSV files help with spreadsheet checks and further analysis. PDF files are useful for reports, notes, client summaries, and saved calculation records.