# Poisson Equation with deal.II + Ginkgo

Solves the Laplace equation $-\Delta u = 1$ on $[-1,1]^2$ with homogeneous
Dirichlet boundary conditions using deal.II for mesh management and assembly,
and Ginkgo for the linear solve.

Based on [deal.II tutorial step-3](https://www.dealii.org/current/doxygen/deal.II/step_3.html).

## Integration Overview

deal.II assembles the system matrix in its own `SparseMatrix` format. We
extract the CSR structure by iterating over rows, then wrap the data into
a `gko::matrix::Csr` using non-owning array views. deal.II vectors are
similarly wrapped as `gko::matrix::Dense` with zero-copy.

## Step-by-Step

### 1. Create the Mesh

```{literalinclude} poisson.cpp
:language: cpp
:start-after: [create-mesh]
:end-before: [/create-mesh]
```

### 2. Set Up Finite Elements

```{literalinclude} poisson.cpp
:language: cpp
:start-after: [setup-fe]
:end-before: [/setup-fe]
```

### 3. Assemble the Linear System

deal.II assembles the stiffness matrix and right-hand side using cell-wise
quadrature, then applies boundary conditions.

```{literalinclude} poisson.cpp
:language: cpp
:start-after: [assemble-system]
:end-before: [/assemble-system]
```

### 4. Extract CSR Data for Ginkgo

deal.II does not expose raw CSR pointers, so we iterate over the matrix rows
to build standard CSR arrays.

```{literalinclude} poisson.cpp
:language: cpp
:start-after: [convert-to-ginkgo]
:end-before: [/convert-to-ginkgo]
```

### 5. Set Up the Ginkgo Executor

```{literalinclude} poisson.cpp
:language: cpp
:start-after: [setup-ginkgo]
:end-before: [/setup-ginkgo]
```

### 6. Wrap as Ginkgo Objects

The CSR arrays and deal.II vectors are wrapped as Ginkgo objects using
non-owning views — no data is copied.

```{literalinclude} poisson.cpp
:language: cpp
:start-after: [create-ginkgo-objects]
:end-before: [/create-ginkgo-objects]
```

### 7. Solve with Ginkgo

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

## Building

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

To specify library paths:

```bash
cmake -S . -B build --preset default \
  -DGinkgo_DIR=/path/to/ginkgo/lib/cmake/Ginkgo \
  -DDEAL_II_DIR=/path/to/dealii
```

## Expected Output

```text
Number of active cells: 1024
Number of degrees of freedom: 1089
Solver converged.
Output written to solution.vtk
```