# Solving PDEs with different scalar (float) types

This demo 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 numpy as np
import scipy.sparse
import scipy.sparse.linalg

import ufl
from dolfinx import common, fem, mesh, plot

from mpi4py import MPI

SciPy solvers do no support MPI, so all computations are performed on a single MPI rank

comm = MPI.COMM_SELF

Create a mesh and function space

msh = mesh.create_rectangle(comm=comm, points=((0.0, 0.0), (2.0, 1.0)), n=(32, 16),
cell_type=mesh.CellType.triangle)
V = fem.FunctionSpace(msh, ("Lagrange", 1))
# Define a variartional problem
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
facets = mesh.locate_entities_boundary(msh, dim=1,
marker=lambda x: np.logical_or(np.isclose(x[0], 0.0),
np.isclose(x[0], 2.0)))
dofs = fem.locate_dofs_topological(V=V, entity_dim=1, entities=facets)
def solve(dtype=np.float32):
"""Solve the variational problem"""

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

# Create a Scipy sparse matrix that shares data with A
As = scipy.sparse.csr_matrix((A.data, A.indices, A.indptr))

# Solve the variational problem and return the solution
uh = fem.Function(V, dtype=dtype)
uh.x.array[:] = scipy.sparse.linalg.spsolve(As, b.array)
return uh
def display(u, filter=np.real):
"""Plot the solution using pyvista"""
try:
import pyvista
cells, types, x = plot.create_vtk_mesh(V)
grid = pyvista.UnstructuredGrid(cells, types, x)
grid.point_data["u"] = filter(u.x.array)
grid.set_active_scalars("u")

plotter = pyvista.Plotter()