Spatial and temporal resolution

Spatial resolution

\(h\)-refinement refines the mesh (locally or globally) to achieve a smaller cell size \(h\) where desired.

\(p\)-refinement increases the polynomial degree \(p\) (locally or globally) of the finite element basis functions. However dolfinx has no support for local \(p\)-refinement.

\(hp\)-refinement combines both strategies.

Method of manufactured solutions

\[\begin{split} \begin{align*} &\text{Find}~u(\textbf{x}): \Omega \to \mathbb{R}~\text{such that} \\ &\mathbb{BVP}_u\begin{cases} \mathscr{L}_{\textbf{x}}(u) = \mathscr{L}_{\textbf{x}}(u_{\text{e}}) & \forall\textbf{x}\in\Omega \\ u=u_{\text{e}} & \forall \textbf{x}\in\partial\Omega \\ \end{cases}~. \end{align*} \end{split}\]

The goal is to demonstrate that the error norm tends to zero as mesh cell cell size tends to zero. Asymptotically as \(h\to0\) for the error \(\mathcal{E}=u_{\text{e}} - u\), we expect

\[||\mathcal{E}||\sim C h^r\]

for some numerical factor \(C>0\) and convergence rate \(r>0\). The \(L_p\) and \(\ell_p\) norms are defined by

\[||\mathcal{E}||_{L_p}=\left(\int_\Omega |\mathcal{E}|^p~\text{d}\Omega\right)^{1/p}\]

\[||\mathcal{E}||_{\ell_p}=\left(\sum_j|E_j|^p\right)^{1/p}\]

if \(\sum_jE_j\xi_j\) is the approximation of \(\mathcal{E}\) in the finite element function space spanned by basis functions \(\{\xi_j\}_j\).

Temporal resolution

\(\Delta t\)-refinement uses a smaller constant time-step.

\(\Delta t\)-adaptivity computes an adaptive time-step based on constraints such as the CFL condition.

Adaptive time-steps for the advection-diffusion-reaction equation

The precise value of Courant number necessary for stability can depend on the choice of discretization in space and time, as well as the number of spatial dimensions.

\[\begin{split} \begin{align*} \Delta t &= \min\{C_{\textbf{u}}\Delta t_{\textbf{u}}, C_{\mathsf{D}}\Delta t_{\mathsf{D}}, C_{R}\Delta t_{R}\} \\ \Delta t_{\textbf{u}} &= \min_{\textbf{x}}\left(\frac{h}{|\textbf{u}|}\right) \\ \Delta t_{\mathsf{D}} & = \min_{\textbf{x}}\left(\frac{h^2}{2|\mathsf{D}|}\right) \\ \Delta t_{R} &= \min_{\textbf{x}}\left(\frac{1}{|R|}\right) \\ \end{align*} \end{split}\]

Method of manufactured solutions

\[\begin{split} \begin{align*} &\text{Find}~u(\textbf{x}, t): \Omega\times[0,\infty) \to \mathbb{R}~\text{such that} \\ &\begin{cases} \frac{\partial u}{\partial t} = \mathscr{L}_{\textbf{x},t}(u) & \forall(\textbf{x}, t)\in\Omega\times[0,\infty) \\ u=u_{\text{e}} & \forall(\textbf{x},t)\in\Omega\times\{0\}\\ u=u_{\text{e}} & \forall(\textbf{x},t)\in\partial\Omega\times[0,\infty) \end{cases}~. \end{align*} \end{split}\]

Given a fixed time-step \(\Delta t\) and mesh cell size \(h\), the error norm at time \(t\) is expected to scale as

\[||\mathcal{E}(t)||\sim C_{\Delta t}\Delta t^{\,r_{\Delta t}} + C_hh^{r_h}\]

with numerical factors \(C_{\Delta t}, C_h\) convergence rates \(r_\Delta, r_h\) themselves being time-dependent. The maximum-in-time error norm is given by

\[\max_{t\geq0}||\mathcal{E}(t)|| = \max_{n\geq0}||\mathcal{E}(n\Delta t)||\]