A nonlinear Poisson equation

Authors: Anders Logg and Hans Petter Langtangen

We shall now address how to solve nonlinear PDEs. We will see that nonlinear problems introduce some subtle differences in how we define the variational form.

The PDE problem

As a model for the solution of nonlinear PDEs, we take the following nonlinear Poisson equation

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

The coefficients \(q(u)\) make the problem nonlinear (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 vanish 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 nonlinear 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 nonlinear in \(u\), the variational statement gives rise to a system of nonlinear algebraic equations in the unknowns \(U_1,\dots,U_N\).