---
tag: 6W72QW363B
kind: concept
name: Method of fundamental solutions
msc: [65N80, 65N35]
sources:
  - url: https://en.wikipedia.org/wiki/Method_of_fundamental_solutions
same_as:
  wikidata: Q6823714
  wikipedia: Method of fundamental solutions
---

## Summary

The method of fundamental solutions approximates the solution of a linear homogeneous elliptic equation by a sum of [[fundamental solutions|K2NK9A8M5F]].
These are centred at source points outside the domain.
Each term solves the equation exactly inside the domain, so only the boundary condition has to be fitted.
It needs neither a mesh nor integrals, only the evaluation of the fundamental solution at boundary points.
It is used when the fundamental solution is known in closed form.
Examples are the Laplace equation $\Delta u = 0$, with $\Delta$ the [[Laplacian|0E0B6RA8K0]], the [[Helmholtz equation|MNXKTXJ35G]], and linear elasticity.
See the [[definition|XS0Q09SV2T]].

## Explanations

### L1 human

Place a few point charges outside a region.
The electric potential they create satisfies the Laplace equation everywhere inside the region.
By choosing the strengths of the charges well, one can make this potential match given values on the boundary of the region, at least approximately.
The method of fundamental solutions does exactly this, and the resulting sum of charge potentials is the approximate solution.