Solve PDEs that model direct current electrical conduction
or other elliptic problems

The direct current conduction problems, such as electrolysis
and computation of resistances of grounding plates, involve a steady
current passing through a conductive medium. Current density **J** is related to the electric field **E** as **J** = *σ***E**, where *σ* is the
conductivity of the medium. Combining the continuity equation ∇ · **J** = *Q*, where *Q* is
the current source, with the definition of the electric potential *V* yields
the elliptic Poisson's equation:

–∇ · (*σ*∇*V*)
= *Q*.

Use the Dirichlet boundary condition to assign values of the
electric potential *V* to the boundaries. Typically,
the boundaries are metallic conductors. Use the Neumann boundary conditions
if the value of the normal component of the current density (**n** · (*σ*∇*V*))
is known. You also can specify a generalized Neumann condition defined
by **n** · (*σ*∇*V*) + *qV* = *g*, where *q* is interpreted
as film conductance for thin plates. For details, see Conductive Media DC.

PDE | Solve partial differential equations in 2-D regions |

Perform one-level domain decomposition for the L-shaped membrane.

Solve a nonlinear problem on a unit disk using the PDE app and the command line.

Solve the Laplace equation for a geometry consisting of two circular metallic conductors placed on a plane.

**Solve Poisson's Equation on a Unit Disk**

Solve a simple elliptic PDE in the form of Poisson's equation on a unit disk.

Solve a nonlinear problem on a unit disk using the PDE app and the command line.

**Solve Poisson's Equation on a Grid**

Description of Partial Differential Equation Toolbox™ solution to Poisson's equation.

Mathematical definition and discussion of the elliptic equation

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