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Problem statement

Problem statement#

The first problem we will encounter in this tutorial is an approximation problem. Given to function \(f\) and \(g\), compute the approximation of \(\frac{f}{g}\) in a specified finite element space. Mathematically, we can write@ Find \(u\in V(\Omega)\) such that

(1)#\[\begin{align} u &= \frac{f(x,y,z)}{g(x,y,z)} \qquad \text{in } \Omega\subset \mathbb{R}^3. \end{align}\]

We will first show how we can easily solve this in FEniCSx on a unit square. Then, we will go through what happens under the hood in the various components listed on the front page. To solve this problem, we create the variational form

(2)#\[\begin{align}\int_\Omega uv~\mathrm{d}x = \int_\Omega \frac{f}{g}v~\mathrm{d}x\quad \forall v \in V\end{align}\]

Reminder: DOLFINx syntax#

As seen in previous classes and tutorials, you can solve this problem in DOLFINx for a 3rd order discontinuous Lagrange element with the following code

import pyvista
import dolfinx.fem.petsc
import dolfinx
from mpi4py import MPI
import ufl
import numpy as np

mesh = dolfinx.mesh.create_unit_square(MPI.COMM_WORLD, 10, 10)
V = dolfinx.fem.FunctionSpace(mesh, ("Discontinuous Lagrange", 3))
u = ufl.TrialFunction(V)
v = ufl.TestFunction(V)
a = u * v * ufl.dx

F = dolfinx.fem.FunctionSpace(mesh, ("Discontinuous Lagrange", 2))
f = dolfinx.fem.Function(F)
G = dolfinx.fem.FunctionSpace(mesh, ("Lagrange", 2))
g = dolfinx.fem.Function(G)

L = f / g * v * ufl.dx

f.interpolate(lambda x: x[0])
g.interpolate(lambda x: 2+np.sin(x[1]))

problem = dolfinx.fem.petsc.LinearProblem(a, L)
uh = problem.solve()

def plot_scalar_function(u: dolfinx.fem.Function):
    pyvista.start_xvfb(1.0)
    u_grid = pyvista.UnstructuredGrid(
        *dolfinx.plot.vtk_mesh(u.function_space))
    u_grid.point_data["u"] = u.x.array
    plotter = pyvista.Plotter()
    plotter.add_mesh(u_grid)
    plotter.show_axes()
    plotter.view_xy()
    plotter.show()

plot_scalar_function(uh)
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