Simple example

Authors: Andrey Latyshev (University of Luxembourg, Sorbonne Université, andrey.latyshev@uni.lu)

In order to show how an external operator can be used in a variational setting in FEniCSx/DOLFINx, we want to start with a simple example.

Let us denote an external operator that is not expressible through UFL by \(N = N(u)\), where the function \(u\) is its single operand from functional space \(V\). To fix ideas, we consider the following linear form \(F\):

\[ F(N(u);v) = \int N(u)v \, dx \]

More generally, we can think of \(F(N, v)\) as a UFL-expressible (non)linear form of the function \(N\). However, as a nonlinear expression of \(u\), the form can not be expressed via UFL.

In a variational setting, we quite often need to compute the Jacobian of the Form \(F\). In other words, we need to take the Gateau derivative of the functional \(F\) in the direction of \(\hat{u}\). Denoting the full and partial Gateau derivatives of a functional through \(\frac{d }{d u}(\cdot)\) and \(\frac{\partial}{\partial u}(\cdot)\) respectively, applying the chain rule and omitting the operand of \(N\), we can express the Jacobian of \(F\) as following:

\[ J(N;\hat{u}, v) = \frac{dF}{du}(N;\hat{u}, v) = \frac{\partial F}{\partial N}(N; \frac{\partial N}{\partial u}(u;\hat{u}), v) = \int \hat{N}(u;\hat{u})v \, dx, \]

where \(\hat{N}(u;\hat{u}) = \frac{\partial N}{\partial u}(u;\hat{u})\) is a Gateau derivative of the external operator \(N\) in the direction of \(\hat{u}\).

Thus, the Jacobian \(J\) involves the computation of \( \frac{\partial N}{\partial u}\) which can be seen as another external operator.

The behaviour of both external operators \(N\) and \(\frac{\partial N}{\partial u}\) must be defined by a user via any callable Python function.