# Helmholtz equation

Helmholtz problem in both complex and real modes In the complex mode, the exact solution is a plane wave propagating at an angle theta to the positive x-axis. Chosen for comparison with results from Ihlenburg’s book “Finite Element Analysis of Acoustic Scattering” p138-139. In real mode, the Method of Manufactured Solutions is used to produce the exact solution and source term.

import numpy as np

import ufl
from dolfinx.fem import Function, FunctionSpace, assemble_scalar, form
from dolfinx.fem.petsc import LinearProblem
from dolfinx.io import XDMFFile
from dolfinx.mesh import create_unit_square
from ufl import dx, grad, inner

from mpi4py import MPI
from petsc4py import PETSc

# wavenumber
k0 = 4 * np.pi

# approximation space polynomial degree
deg = 1

# number of elements in each direction of msh
n_elem = 128

msh = create_unit_square(MPI.COMM_WORLD, n_elem, n_elem)
n = ufl.FacetNormal(msh)

# Source amplitude
if np.issubdtype(PETSc.ScalarType, np.complexfloating):
A = PETSc.ScalarType(1 + 1j)
else:
A = 1

# Test and trial function space
V = FunctionSpace(msh, ("Lagrange", deg))

# Define variational problem
u = ufl.TrialFunction(V)
v = ufl.TestFunction(V)
f = Function(V)
f.interpolate(lambda x: A * k0**2 * np.cos(k0 * x) * np.cos(k0 * x))
a = inner(grad(u), grad(v)) * dx - k0**2 * inner(u, v) * dx
L = inner(f, v) * dx

# Compute solution
uh = Function(V)
uh.name = "u"
problem = LinearProblem(a, L, u=uh, petsc_options={"ksp_type": "preonly", "pc_type": "lu"})
problem.solve()

# Save solution in XDMF format (to be viewed in ParaView, for example)
with XDMFFile(MPI.COMM_WORLD, "out_helmholtz/plane_wave.xdmf", "w", encoding=XDMFFile.Encoding.HDF5) as file:
file.write_mesh(msh)
file.write_function(uh)


Calculate $$L_2$$ and $$H^1$$ errors of FEM solution and best approximation. This demonstrates the error bounds given in Ihlenburg. Pollution errors are evident for high wavenumbers.

# Function space for exact solution - need it to be higher than deg
V_exact = FunctionSpace(msh, ("Lagrange", deg + 3))
u_exact = Function(V_exact)
u_exact.interpolate(lambda x: A * np.cos(k0 * x) * np.cos(k0 * x))

# H1 errors
diff = uh - u_exact