A nonlinear Poisson equation

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

(28)\[\begin{align} - \nabla \cdot (q(u) \nabla u)&=f && \text{in } \Omega,\\ u&=u_D && \text{on } \partial \Omega, \end{align}\]

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

(29)\[\begin{align} F(u; v)&=0 && \forall v \in \hat{V}, \end{align}\]

where

(30)\[\begin{align} F(u; v)&=\int_{\Omega}(q(u)\nabla u \cdot \nabla v - fv)\mathrm{d}x, \end{align}\]

and

(31)\[\begin{align} V&=\left\{v\in H^1(\Omega)\vert v=u_D \text{on } \partial \Omega \right\}\\ \hat{V}&=\left\{v\in H^1(\Omega)\vert v=0 \text{on } \partial \Omega \right\} \end{align}\]

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

(32)\[\begin{align} F(u_h, v) &=0 \quad \forall v \in \hat{V}_h, \end{align}\]

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 \(U_1,\dots,U_N\).