# High-Order Diffusion with MFEM + Ginkgo

Solves $-\nabla \cdot (\kappa \nabla u) = 1$ with a spatially varying
coefficient $\kappa(x) = 1 + x_0^2$ using quartic ($p=4$) finite elements.
Shows two integration approaches: manual CSR wrapping and MFEM's built-in
Ginkgo interface.

## Integration Overview

High-order elements produce denser matrix rows (more non-zeros per row),
making solver and preconditioner choice more impactful. This example
demonstrates both integration paths:

1. **Manual CSR wrapping** — full control over Ginkgo's solver factory,
   preconditioner, and stopping criteria
2. **MFEM built-in** (requires `MFEM_USE_GINKGO=ON`) — simpler but
   less configurable

## Step-by-Step

### Variable Coefficient

```{literalinclude} high_order.cpp
:language: cpp
:start-after: [variable-coefficient]
:end-before: [/variable-coefficient]
```

### Assembly with High-Order Elements

The only change from linear elements is `order = 4` in the `H1_FECollection`.
The assembled matrix is larger and denser.

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

### Approach 1: Manual Ginkgo CSR Wrapping

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

### Approach 2: MFEM Built-in Ginkgo Interface

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

## Building

```bash
cmake -S . -B build --preset default
cmake --build build
./build/high_order -o 4 -r 3
```

## Expected Output

```text
Options used:
   --order 4
   --refine 2
Order: 4
Elements: 1024
DOFs: 16641
System: 16641 x 16641, nnz: 591361 (avg nnz/row: 35)

Solving with manual Ginkgo CSR wrapping...
Converged.
Output written to solution.gf and mesh.mesh
```