# Documentation

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# DC Conduction and Elliptic Problems

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 = σE, 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:

–∇ · (σV) = 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 · (σV)) is known. You also can specify a generalized Neumann condition defined by n · (σV) + qV = g, where q is interpreted as film conductance for thin plates. For details, see Conductive Media DC.

## Apps

 PDE Modeler Solve partial differential equations in 2-D regions

## Topics

### Programmatic Workflow

Solve Poisson's Equation on a Unit Disk

Use command-line functions to solve a simple elliptic PDE in the form of Poisson's equation on a unit disk.

Domain Decomposition Problem

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

Minimal Surface Problem

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

### PDE Modeler App Workflow

Solve Poisson's Equation on a Unit Disk: PDE Modeler App

Use the PDE Modeler app to solve a simple elliptic PDE in the form of Poisson's equation on a unit disk.

Conductive Media DC

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

Minimal Surface Problem

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

### Agorithms and Other Theory

Solve Poisson's Equation on a Grid

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

Elliptic Equations

Mathematical definition and discussion of the elliptic equation