# A nonlinear Poisson equation

## Contents

# A nonlinear Poisson equation¶

Authors: Anders Logg and Hans Petter Langtangen

We shall now address how to solve non-linear PDEs. We will see that non-linear problems introduce some subtle differences on how we define the variational form.

## The PDE problem¶

As a model for the solution of non-linear PDEs, we take the following non-linear Poisson equation

and the coefficients \(q(u)\) makes the problem non-linear (unless q(u) is constant in \(u\)).

## Variational formulation¶

As usual, we multiply the PDE by a test function \(v\in \hat{V}\), integrate over the domain, and integrate second-order derivatives by parts. The boundary integrals arising from integration by parts vanishes wherever we employ Dirichlet conditions. The resulting variational formulation of our model problem becomes:

Find \(u\in V\) such that

where

and

The discrete problem arises as usual by restricting \(V\) and \(\hat{V}\) to a pair of discrete spaces. The discrete non-linear problem can therefore be written as:

Find \(u_h \in V_h\) such that

with \(u_h=\sum_{j=1}^N U_j\phi_j\). Since \(F\) is non-linear in \(u\), the variational statement gives rise to a system of non-linear algebraic equation in the unknowns \(U_1,\dots,U_N\).