# Poisson Equation with MFEM + Ginkgo

Solves the Laplace equation $-\Delta u = 1$ with homogeneous Dirichlet
boundary conditions using MFEM for finite element assembly and Ginkgo
for the linear solve.

Based on [MFEM Example 1](https://mfem.org/examples/).

## Integration Overview

MFEM's `SparseMatrix` stores CSR data directly. After calling `Finalize()`,
the raw arrays are accessible via `GetI()`, `GetJ()`, and `GetData()`. These
are wrapped into a `gko::matrix::Csr` using `gko::array::view` for zero-copy
interoperability.

## Step-by-Step

### 1. Set Up Mesh and Finite Element Space

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

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

### 2. Assemble the Linear System

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

### 3. Convert to Ginkgo Objects

MFEM exposes CSR pointers directly — `GetI()` returns row pointers, `GetJ()`
returns column indices, `GetData()` returns values.

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

### 4. Solve with Ginkgo

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

### 5. Recover Solution

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

## Building

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

A mesh file can be passed with `-m`:

```bash
./build/poisson -m /path/to/mfem/data/star.mesh -o 2
```

## Expected Output

```text
Options used:
   --mesh
   --order 1
   --refine 2
Number of elements: 4096
Number of unknowns: 4225
System size: 4225 x 4225, nnz: 37249
Ginkgo solver converged.
Output written to solution.gf and mesh.mesh
```