Adaptivity
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Basic concept
The adaptive methodology in this paper behaves similarly to "remeshing" used commonly in FEM. Some initial (possibly variable) nodal spacing $h^0$ is chosen, as well as its lower and upper bounds $h_L$ and $h_U$, respectively. 3 Domain $\Omega$ is filled with nodes, conforming to $h^0$ and the solution $u^0$ is obtained. An error indicator is employed to determine which nodes should be (de)refined and the nodal density $h^0$ is altered appropriately. This adaptive cycle below is repeated until the convergence criterion is met. The procedure on $j$-th iteration is written in more detail below:
- Fill $\Omega$ with nodes conforming to $h^j$.
- Solve the problem to obtain $u^j$.
- Compute the error indicator values $\varepsilon_i^j$ for each node $p_i$.
- If the mean of $\varepsilon_i^j$ is below some tolerance $\varepsilon$ return $u^j$ as the solution and stop.
- Adapt $h^j$ to obtain $h^{j+1}$.
More details can be found in our paper: https://arxiv.org/abs/1811.10368
Node density adaptation
The existing nodal spacing function $h^j$ is evaluated at nodes $p_i$ to obtain values $h_{i,j} = h^j(p_i)$ . These values $h_{i,j}$ are modified by a density factor $f_i$ as $$ h_{i,j+1} = \min(\max(h_{i,j} / f_i, h_L(p_i)), h_U(p_i)), $$ where density factor $f_i$ is computed as $$ f_i = \begin{cases} 1 + \frac{\eta - \varepsilon_i}{\eta - m} (\frac{1}{\beta} - 1), & \varepsilon_i \leq \eta, \quad \text{i.e. decrease the density} \\ 1, & \eta < \varepsilon_i < \varepsilon, \quad \text{i.e. no change in density}\\ 1 + \frac{\varepsilon_i - \varepsilon}{M - \varepsilon} (\alpha - 1), & \varepsilon_i \geq \varepsilon, \quad \text{i.e. increase the density} \end{cases} $$ and $\alpha$ represents the refine aggressiveness, $\beta$ the derefine aggressiveness, $\varepsilon$ the refinement threshold, $\eta$ the derenfinement threshold, and $m = \min_i \varepsilon_i$ is the minimal and $M = \max_i \varepsilon_i$ is the maximal value of the error indicator. Note that setting $\alpha=1$ or $\beta=1$ disables refinement and derefinement, respectively.
Error indicators
The work on error indicators is ongoing. For now, we use an ad hoc error indicator $$\varepsilon_i = std_{j \in I_i}(u_j)$$, which represents the standard deviation of function values over all stencil nodes of a given node $p_i$.
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