In recent years, there has been an explosion in the number of articles on fractional derivatives, which are derivatives with a non-integer order.
In this talk, I will provide an introduction to this topic and consider the subdiffusion problem
\begin{align*}
\partial_t^{\alpha}u + L u&=f(x,t)&&\text{for }(x,t)\in\Omega\times(0,T],
\end{align*}
where $\alpha\in(0,1)$, $\partial_t^\alpha$ is the Caputo fractional derivative operator and $L$ is a second-order elliptic spatial operator.
The problem is subject to an initial condition, $u(\cdot,0)=u_0$, and homogeneous boundary conditions in space.
We will approximate this problem numerically using a collocation method in time and a mesh adaptation process with an a-posteriori error estimator, employing piecewise polynomials of arbitrary order.
We will also consider the existence of a discrete solution to the problem.
This is a joint work with Natalia Kopteva (University of Limerick, Ireland).