# Adaptive Refinement with deal.II + Ginkgo

Solves the Poisson equation $-\Delta u = 1$ on a disk with adaptive mesh
refinement (AMR). After each refinement cycle, the mesh changes, the system
is re-assembled, and a **new Ginkgo solver is generated** from the updated
matrix.

Based on [deal.II tutorial step-6](https://www.dealii.org/current/doxygen/deal.II/step_6.html).

## Integration Overview

The key difference from the basic Poisson example is that AMR changes the
matrix dimensions and sparsity pattern every cycle. The Ginkgo solver (and
its preconditioner) must be re-created from scratch after each refinement.
This example shows the pattern for solver re-generation.

## Step-by-Step

### CSR Extraction Helper

A reusable function that extracts CSR arrays from a deal.II `SparseMatrix`:

```{literalinclude} adaptive_refinement.cpp
:language: cpp
:start-after: [helper-extract-csr]
:end-before: [/helper-extract-csr]
```

### Solve Function with Ginkgo

Each cycle calls this function, which creates a fresh Ginkgo solver.
After solving, `constraints.distribute()` enforces hanging node constraints.

```{literalinclude} adaptive_refinement.cpp
:language: cpp
:start-after: [solve-with-ginkgo]
:end-before: [/solve-with-ginkgo]
```

### The Refinement Loop

The main loop: generate/refine mesh, set up DOFs, assemble, solve, repeat.

```{literalinclude} adaptive_refinement.cpp
:language: cpp
:start-after: [refinement-loop]
:end-before: [/refinement-loop]
```

## Building

```bash
cmake -S . -B build --preset default
cmake --build build
./build/adaptive_refinement
```

## Expected Output

```text
Cycle 0:
  Cells: 20  DOFs: 89
  Solved.
Cycle 1:
  Cells: 44  DOFs: 209
  Solved.
...
Cycle 7:
  Cells: 2513  DOFs: 12725
  Solved.
```

Each cycle produces a `solution-N.vtk` file for visualization.