Laplace Equation Solver Calculator

Example Data Table

Formula Used

The two-dimensional Laplace equation is d2u/dx2 + d2u/dy2 = 0, representing harmonic fields such as steady temperature or electrostatic potential in source-free regions.

With a uniform grid, central differences yield the discrete update: u(i,j) = 0.25 [u(i-1,j)+u(i+1,j)+u(i,j-1)+u(i,j+1)].

The solver iterates until the maximum node change falls below tolerance, or the iteration limit is reached.

How to Use This Calculator

1. Set Lx and Ly for your rectangle.
2. Choose Nx and Ny for resolution.
3. Pick constant edges or corner-based linear edges.
4. Select a method; use SOR omega for speed.
5. Enter tolerance and maximum iterations, then solve.
6. Download CSV or PDF to keep your results.

Professional Article

1) Purpose of the solver

This calculator estimates a harmonic field inside a rectangular region where internal sources are absent. Typical interpretations include steady temperature in a solid without heat generation, electrostatic potential in a charge-free volume, or velocity potential for incompressible, irrotational flow.

2) Governing relationship

The model satisfies Laplace’s equation, meaning the curvature in one direction balances the curvature in the other. Physically, the interior adjusts until no net “driving” remains. This is why boundary values fully determine the interior profile and why solutions are smooth.

3) Grid and discretization

The domain is divided into Nx by Ny nodes with uniform spacing. Central differences approximate second derivatives, producing an averaging rule at each interior node. Increasing grid density generally improves spatial resolution, but it raises computation cost and can require more iterations to converge.

4) Boundary condition options

Two boundary styles are supported. Constant edges apply fixed values on top, bottom, left, and right. Corner-based linear edges interpolate along each side between corner values, providing a simple way to model gradual boundary variation when only corner measurements are available.

5) Iterative methods and performance

Jacobi updates all interior nodes from the previous iteration, making it easy to understand but typically slower. Gauss–Seidel updates in-place and often converges faster. Successive over-relaxation (SOR) accelerates Gauss–Seidel by blending the new estimate with the prior value.

6) Selecting relaxation omega

SOR uses omega between 1.0 and 1.99. Values around 1.6 to 1.8 often reduce iteration counts on many grids. If omega is too large, the solution may converge slowly or show oscillatory behavior. Lower omega improves stability at the expense of speed.

7) Convergence and quality checks

This tool stops when the maximum absolute node update falls below the tolerance or when the iteration limit is reached. For confidence, rerun with a tighter tolerance and verify that key values change minimally. Also refine the grid to assess discretization sensitivity and symmetry preservation.

8) Reporting and real-world use

CSV export provides node indices, coordinates, and field values for plotting, documentation, or downstream analysis. The PDF summary captures method settings and basic statistics for audit trails. Common uses include insulation studies, electrode shaping concepts, and educational demonstrations of uniqueness and maximum principles.

FAQs

1) What problem type does this solve?

It solves a steady, source-free field on a rectangle using finite differences. Examples include equilibrium heat flow without generation, electrostatic potential in charge-free regions, and potential-flow models in fluids.

2) How should I choose Nx and Ny?

Start with 21 x 21 to explore trends, then increase resolution to confirm stability of mid-domain values. If results change noticeably, refine further until changes are acceptably small for your application.

3) When is corner-based boundary interpolation useful?

Use it when you only know corner values or expect edges to vary smoothly. It creates linear variation along each side, producing a more realistic boundary than constant edges for gradual transitions.

4) Which iterative method is best?

SOR is usually fastest. Gauss–Seidel is a solid default without tuning. Jacobi is simplest for learning and comparison, but it often needs more iterations for the same tolerance.

5) What omega value should I try for SOR?

Try 1.6 to 1.8 first. If convergence slows or values appear unstable, reduce omega. If convergence is stable but slow, adjust omega slightly upward in small steps.

6) What tolerance is practical?

For visualization or early design checks, 1e-4 to 1e-5 is commonly adequate. For tighter comparisons or reports, use 1e-6 or smaller and confirm results with a refined grid.

7) Why is the sampled u(x,y) “nearest node”?

Nearest-node sampling is transparent and fast. For smoother point estimates, increase Nx and Ny or export CSV and apply bilinear interpolation between surrounding nodes during post-processing.