Editing Equations, Formulas and Algorithms (section)

== Finite Element Methods (FEM) ==
Finite element methods divide the domain of a differential equation into smaller elements, allowing for the approximation of the solution within each element. FEM is employed in structural analysis, electromagnetics, and other fields to solve partial differential equations and boundary value problems.

'''Example: Finite Element Method'''
Consider the one-dimensional Poisson equation:
<math mode="display"> -\frac{d^2 u}{dx^2} = f(x) </math>
where <math> u(x) </math> is the unknown function to be solved for and <math> f(x) </math> is a given function.

'''Source Code Example (Python with FEniCS):'''
<syntaxhighlight lang="python">
from fenics import *

# Define mesh
mesh = IntervalMesh(100, 0, 1)

# Define function space
V = FunctionSpace(mesh, 'CG', 1)

# Define boundary conditions
u_L = Constant(0.0)
u_R = Constant(1.0)
bc_L = DirichletBC(V, u_L, 'on_boundary && near(x[0], 0)')
bc_R = DirichletBC(V, u_R, 'on_boundary && near(x[0], 1)')
bcs = [bc_L, bc_R]

# Define variational problem
u = TrialFunction(V)
v = TestFunction(V)
f = Constant(1.0)
a = inner(grad(u), grad(v)) * dx
L = f * v * dx

# Solve variational problem
u = Function(V)
solve(a == L, u, bcs)

# Plot solution
plot(u)
plt.show()
</syntaxhighlight>