How to solve the Poisson equation in FEniCS?
Solving a boundary-value problem such as the Poisson equation in FEniCS consists of the following steps: Identify the computational domain (\\(\\Omega\\)), the PDE, its boundary conditions, and source terms (\\(f\\)). Reformulate the PDE as a finite element variational problem.
How to solve the Poisson equation in Python?
Write a Python program which defines the computational domain, the variational problem, the boundary conditions, and source terms, using the corresponding FEniCS abstractions. Call FEniCS to solve the boundary-value problem and, optionally, extend the program to compute derived quantities such as fluxes and averages, and visualize the results.
How to write out the Poisson equation in two dimensions?
In two space dimensions with coordinates x and y, we can write out the Poisson equation as − ∂2u ∂x2 − ∂2u ∂y2 = f(x, y). The unknown u is now a function of two variables, u = u(x, y), defined over a two-dimensional domain Ω.
Which is the bilinear form of the Poisson equation?
For the Poisson equation, we have: L(v) = ∫Ωfvdx. From the mathematics literature, a(u, v) is known as a bilinear form and L(v) as a linear form. We shall, in every linear problem we solve, identify the terms with the unknown u and collect them in a(u, v), and similarly collect all terms with only known functions in L(v).
Is there a solver for the nonlinear Poisson equation?
A solver for the nonlinear Poisson equation is as easy to implement as a solver for the linear Poisson equation. All we need to do is to state the formula for F and call solve (F == 0, u, bc) instead of solve (a == L, u, bc) as we did in the linear case. Here is a minimalistic code:
This demo is implemented in a single Python file, demo_poisson.py, which contains both the variational forms and the solver. This demo illustrates how to: The solution for u in this demo will look as follows: 16.1. Equation and problem definition The Poisson equation is the canonical elliptic partial differential equation.