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# Solving PDEs with different scalar (float) types

This demo  ({download}`demo_types.py`) shows:

- How to solve problems using different scalar types, .e.g. single or
  double precision, or complex numbers
- Interfacing with [SciPy](https://scipy.org/) sparse linear algebra
  functionality

```python
from mpi4py import MPI
```

```python
import numpy as np
import scipy.sparse
import scipy.sparse.linalg

import ufl
from dolfinx import fem, la, mesh, plot
```

SciPy solvers do not support MPI, so all computations will be
performed on a single MPI rank

```python
comm = MPI.COMM_SELF
```

Create a function that solves the Poisson equation using different
precision float and complex scalar types for the finite element
solution.

```python
def display_scalar(u, name, filter=np.real):
    """Plot the solution using pyvista"""
    try:
        import pyvista

        cells, types, x = plot.vtk_mesh(u.function_space)
        grid = pyvista.UnstructuredGrid(cells, types, x)
        grid.point_data["u"] = filter(u.x.array)
        grid.set_active_scalars("u")
        plotter = pyvista.Plotter()
        plotter.add_mesh(grid, show_edges=True)
        plotter.add_mesh(grid.warp_by_scalar())
        plotter.add_title(f"{name}: real" if filter is np.real else f"{name}: imag")
        if pyvista.OFF_SCREEN:
            pyvista.start_xvfb(wait=0.1)
            plotter.screenshot(f"u_{'real' if filter is np.real else 'imag'}.png")
        else:
            plotter.show()
    except ModuleNotFoundError:
        print("'pyvista' is required to visualise the solution")
```

```python
def display_vector(u, name, filter=np.real):
    """Plot the solution using pyvista"""
    try:
        import pyvista

        V = u.function_space
        cells, types, x = plot.vtk_mesh(V)
        grid = pyvista.UnstructuredGrid(cells, types, x)
        grid.point_data["u"] = filter(
            np.insert(u.x.array.reshape(x.shape[0], V.dofmap.index_map_bs), 2, 0, axis=1)
        )
        plotter = pyvista.Plotter()
        plotter.add_mesh(grid.warp_by_scalar(), show_edges=True)
        plotter.add_title(f"{name}: real" if filter is np.real else f"{name}: imag")
        if pyvista.OFF_SCREEN:
            pyvista.start_xvfb(wait=0.1)
            plotter.screenshot(f"u_{'real' if filter is np.real else 'imag'}.png")
        else:
            plotter.show()
    except ModuleNotFoundError:
        print("'pyvista' is required to visualise the solution")
```

```python
def poisson(dtype):
    """Poisson problem solver

    Args:
        dtype: Scalar type to use.


    """

    # Create a mesh and locate facets by a geometric condition
    msh = mesh.create_rectangle(
        comm=comm,
        points=((0.0, 0.0), (2.0, 1.0)),
        n=(32, 16),
        cell_type=mesh.CellType.triangle,
        dtype=np.real(dtype(0)).dtype,
    )
    facets = mesh.locate_entities_boundary(
        msh, dim=1, marker=lambda x: np.isclose(x[0], 0.0) | np.isclose(x[0], 2.0)
    )

    # Define a variational problem.
    V = fem.functionspace(msh, ("Lagrange", 1))
    u, v = ufl.TrialFunction(V), ufl.TestFunction(V)
    x = ufl.SpatialCoordinate(msh)
    fr = 10 * ufl.exp(-((x[0] - 0.5) ** 2 + (x[1] - 0.5) ** 2) / 0.02)
    fc = ufl.sin(2 * np.pi * x[0]) + 10 * ufl.sin(4 * np.pi * x[1]) * 1j
    gr = ufl.sin(5 * x[0])
    gc = ufl.sin(5 * x[0]) * 1j
    a = ufl.inner(ufl.grad(u), ufl.grad(v)) * ufl.dx
    L = ufl.inner(fr + fc, v) * ufl.dx + ufl.inner(gr + gc, v) * ufl.ds

    # In preparation for constructing Dirichlet boundary conditions, locate
    # facets on the constrained boundary and the corresponding
    # degrees-of-freedom.
    dofs = fem.locate_dofs_topological(V=V, entity_dim=1, entities=facets)

    # Process forms. This will compile the forms for the requested type.
    a0 = fem.form(a, dtype=dtype)
    if np.issubdtype(dtype, np.complexfloating):
        L0 = fem.form(L, dtype=dtype)
    else:
        L0 = fem.form(ufl.replace(L, {fc: 0, gc: 0}), dtype=dtype)

    # Create a Dirichlet boundary condition
    bc = fem.dirichletbc(value=dtype(0), dofs=dofs, V=V)

    # Assemble forms
    A = fem.assemble_matrix(a0, [bc])
    A.scatter_reverse()
    b = fem.assemble_vector(L0)
    fem.apply_lifting(b.array, [a0], bcs=[[bc]])
    b.scatter_reverse(la.InsertMode.add)
    fem.set_bc(b.array, [bc])

    # Create a SciPy CSR  matrix that shares data with A and solve
    As = scipy.sparse.csr_matrix((A.data, A.indices, A.indptr))
    uh = fem.Function(V, dtype=dtype)
    uh.x.array[:] = scipy.sparse.linalg.spsolve(As, b.array)

    return uh
```

Create a function that solves the linearised elasticity equation using
different precision float and complex scalar types for the finite
element solution.

```python
def elasticity(dtype) -> fem.Function:
    """Linearised elasticity problem solver."""

    # Create a mesh and locate facets by a geometric condition
    msh = mesh.create_rectangle(
        comm=comm,
        points=((0.0, 0.0), (2.0, 1.0)),
        n=(32, 16),
        cell_type=mesh.CellType.triangle,
        dtype=np.real(dtype(0)).dtype,
    )
    facets = mesh.locate_entities_boundary(
        msh, dim=1, marker=lambda x: np.isclose(x[0], 0.0) | np.isclose(x[0], 2.0)
    )

    # Define the variational problem.
    gdim = msh.geometry.dim
    V = fem.functionspace(msh, ("Lagrange", 1, (gdim,)))
    ω, ρ = 300.0, 10.0
    x = ufl.SpatialCoordinate(msh)
    f = ufl.as_vector((ρ * ω**2 * x[0], ρ * ω**2 * x[1]))

    E, ν = 1.0e6, 0.3
    μ, λ = E / (2.0 * (1.0 + ν)), E * ν / ((1.0 + ν) * (1.0 - 2.0 * ν))

    def σ(v):
        """Return an expression for the stress σ given a displacement field"""
        return 2.0 * μ * ufl.sym(ufl.grad(v)) + λ * ufl.tr(ufl.sym(ufl.grad(v))) * ufl.Identity(
            len(v)
        )

    u, v = ufl.TrialFunction(V), ufl.TestFunction(V)
    a = ufl.inner(σ(u), ufl.grad(v)) * ufl.dx
    L = ufl.inner(f, v) * ufl.dx

    dofs = fem.locate_dofs_topological(V=V, entity_dim=1, entities=facets)

    # Process forms. This will compile the forms for the requested type.
    a0, L0 = fem.form(a, dtype=dtype), fem.form(L, dtype=dtype)

    # Create a Dirichlet boundary condition
    bc = fem.dirichletbc(np.zeros(2, dtype=dtype), dofs, V=V)

    # Assemble forms (CSR matrix)
    A = fem.assemble_matrix(a0, [bc], block_mode=la.BlockMode.expanded)
    A.scatter_reverse()
    assert A.block_size == [1, 1]

    b = fem.assemble_vector(L0)
    fem.apply_lifting(b.array, [a0], bcs=[[bc]])
    b.scatter_reverse(la.InsertMode.add)
    fem.set_bc(b.array, [bc])

    # Create a SciPy CSR  matrix that shares data with A and solve
    As = scipy.sparse.csr_matrix((A.data, A.indices, A.indptr))
    uh = fem.Function(V, dtype=dtype)
    uh.x.array[:] = scipy.sparse.linalg.spsolve(As, b.array)

    return uh
```

Solve problems for different types

```python
uh = poisson(dtype=np.float32)
uh = poisson(dtype=np.float64)
uh = poisson(dtype=np.complex64)
uh = poisson(dtype=np.complex128)
display_scalar(uh, "poisson", np.real)
display_scalar(uh, "poisson", np.imag)
```

```python
uh = elasticity(dtype=np.float32)
uh = elasticity(dtype=np.float64)
uh = elasticity(dtype=np.complex64)
uh = elasticity(dtype=np.complex128)
display_vector(uh, "elasticity", np.real)
```