# Solving PDEs with different scalar (float) types

This demo (demo_types.py) shows:

• How to solve problems using different scalar types, .e.g. single or double precision, or complex numbers

• Interfacing with SciPy sparse linear algebra functionality

import sys

from mpi4py import MPI

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

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.

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_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")

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)
bs = V.dofmap.index_map_bs
grid.point_data["u"] = filter(np.insert(u.x.array.reshape(x.shape[0], bs), bs, 0, axis=1))
plotter = pyvista.Plotter()
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")

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
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]])
fem.set_bc(b.array, [bc])

# Create a SciPy CSR  matrix that shares data with A and solve
As = A.to_scipy()
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.

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])
A.scatter_reverse()

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

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

return uh


Solve problems for different types

uh = poisson(dtype=np.float32)
uh = poisson(dtype=np.float64)
if not sys.platform.startswith("win32"):
uh = poisson(dtype=np.complex64)
uh = poisson(dtype=np.complex128)
display_scalar(uh, "poisson", np.real)
display_scalar(uh, "poisson", np.imag)

uh = elasticity(dtype=np.float32)
uh = elasticity(dtype=np.float64)
if not sys.platform.startswith("win32"):
uh = elasticity(dtype=np.complex64)
uh = elasticity(dtype=np.complex128)
display_vector(uh, "elasticity", np.real)