Discontinuous Galerkin heat equation

Figure 1: Temperature field on the unit square with an internal uniform heat source solved with inhomogeneous Dirichlet boundary conditions on the left and right boundaries and flux on the top and bottom boundaries.

Tip

This example is also available as a Jupyter notebook: dg_heat_equation.ipynb.

This example was developed as part of the Google summer of code funded project "Discontinuous Galerkin Infrastructure For the finite element toolbox Ferrite.jl"

Introduction

This tutorial extends Tutorial 1: Heat equation by using the discontinuous Galerkin method. The reader is expected to have gone through Tutorial 1: Heat equation before proceeding with this tutorial. The main differences between the two tutorials are the interface integral terms in the weak form, the boundary conditions, and some implementation differences explained in the commented program below.

The strong form considered in this tutorial is given as follows

\[ -\boldsymbol{\nabla} \cdot [\boldsymbol{\nabla}(u)] = 1 \quad \textbf{x} \in \Omega,\]

with the inhomogeneous Dirichlet boundary conditions

\[u(\textbf{x}) = 1 \quad \textbf{x} \in \partial \Omega_D^+ = \lbrace\textbf{x} : x_1 = 1.0\rbrace, \\ u(\textbf{x}) = -1 \quad \textbf{x} \in \partial \Omega_D^- = \lbrace\textbf{x} : x_1 = -1.0\rbrace,\]

and Neumann boundary conditions

\[[\boldsymbol{\nabla} (u(\textbf{x}))] \cdot \boldsymbol{n} = 1 \quad \textbf{x} \in \partial \Omega_N^+ = \lbrace\textbf{x} : x_2 = 1.0\rbrace, \\ [\boldsymbol{\nabla} (u(\textbf{x}))] \cdot \boldsymbol{n} = -1 \quad \textbf{x} \in \partial \Omega_N^- = \lbrace\textbf{x} : x_2 = -1.0\rbrace,\]

The following definitions of average and jump on interfaces between elements are adopted in this tutorial:

\[ \{u\} = \frac{1}{2}(u^+ + u^-),\quad \llbracket u\rrbracket = u^+ \boldsymbol{n}^+ + u^- \boldsymbol{n}^-\\\]

where $u^+$ and $u^-$ are the temperature on the two sides of the interface.

Derivation of the weak form for homogeneous Dirichlet boundary condition

Defining $\boldsymbol{\sigma}$ as the gradient of the temperature field the equation can be expressed as

\[ \boldsymbol{\sigma} = \boldsymbol{\nabla} (u),\\ -\boldsymbol{\nabla} \cdot \boldsymbol{\sigma} = 1,\]

Multiplying by test functions $ \boldsymbol{\tau} $ and $ \delta u $ respectively and integrating over the domain,

\[ \int_\Omega \boldsymbol{\sigma} \cdot \boldsymbol{\tau} \,\mathrm{d}\Omega = \int_\Omega [\boldsymbol{\nabla} (u)] \cdot \boldsymbol{\tau} \,\mathrm{d}\Omega,\\ -\int_\Omega \boldsymbol{\nabla} \cdot \boldsymbol{\sigma} \delta u \,\mathrm{d}\Omega = \int_\Omega \delta u \,\mathrm{d}\Omega,\]

Integrating by parts and applying divergence theorem,

\[ \int_\Omega \boldsymbol{\sigma} \cdot \boldsymbol{\tau} \,\mathrm{d}\Omega = -\int_\Omega u (\boldsymbol{\nabla} \cdot \boldsymbol{\tau}) \,\mathrm{d}\Omega + \int_\Gamma \hat{u} \boldsymbol{\tau} \cdot \boldsymbol{n} \,\mathrm{d}\Gamma,\\ \int_\Omega \boldsymbol{\sigma} \cdot [\boldsymbol{\nabla} (\delta u)] \,\mathrm{d}\Omega = \int_\Omega \delta u \,\mathrm{d}\Omega + \int_\Gamma \delta u \boldsymbol{\hat{\sigma}} \cdot \boldsymbol{n} \,\mathrm{d}\Gamma,\]

Where $\boldsymbol{n}$ is the outwards pointing normal, $\Gamma$ is the union of the elements' boundaries, and $\hat{u}, \, \hat{\sigma}$ are the numerical fluxes. Substituting the integrals of form

\[ \int_\Gamma q \boldsymbol{\phi} \cdot \boldsymbol{n} \,\mathrm{d}\Gamma = \int_\Gamma \llbracket q\rrbracket \cdot \{\boldsymbol{\phi}\} \,\mathrm{d}\Gamma + \int_{\Gamma^0} \{q\} \llbracket \boldsymbol{\phi}\rrbracket \,\mathrm{d}\Gamma^0,\]

where $\Gamma^0 : \Gamma \setminus \partial \Omega$, and the jump of the vector-valued field $\boldsymbol{\phi}$ is defined as

\[ \llbracket \boldsymbol{\phi}\rrbracket = \boldsymbol{\phi}^+ \cdot \boldsymbol{n}^+ + \boldsymbol{\phi}^- \cdot \boldsymbol{n}^-\\\]

with the jumps and averages results in

\[ \int_\Omega \boldsymbol{\sigma} \cdot \boldsymbol{\tau} \,\mathrm{d}\Omega = -\int_\Omega u (\boldsymbol{\nabla} \cdot \boldsymbol{\tau}) \,\mathrm{d}\Omega + \int_\Gamma \llbracket \hat{u}\rrbracket \cdot \{\boldsymbol{\tau}\} \,\mathrm{d}\Gamma + \int_{\Gamma^0} \{\hat{u}\} \llbracket \boldsymbol{\tau}\rrbracket \,\mathrm{d}\Gamma^0,\\ \int_\Omega \boldsymbol{\sigma} \cdot [\boldsymbol{\nabla} (\delta u)] \,\mathrm{d}\Omega = \int_\Omega \delta u \,\mathrm{d}\Omega + \int_\Gamma \llbracket \delta u\rrbracket \cdot \{\hat{\boldsymbol{\sigma}}\} \,\mathrm{d}\Gamma + \int_{\Gamma^0} \{\delta u\} \llbracket \hat{\boldsymbol{\sigma}}\rrbracket \,\mathrm{d}\Gamma^0,\]

Integrating $ \int_\Omega [\boldsymbol{\nabla} (u)] \cdot \boldsymbol{\tau} \,\mathrm{d}\Omega $ by parts and applying divergence theorem without using numerical flux, then substitute in the equation to obtain a weak form.

\[ \int_\Omega \boldsymbol{\sigma} \cdot \boldsymbol{\tau} \,\mathrm{d}\Omega = \int_\Omega [\boldsymbol{\nabla} (u)] \cdot \boldsymbol{\tau} \,\mathrm{d}\Omega + \int_\Gamma \llbracket \hat{u} - u\rrbracket \cdot \{\boldsymbol{\tau}\} \,\mathrm{d}\Gamma + \int_{\Gamma^0} \{\hat{u} - u\} \llbracket \boldsymbol{\tau}\rrbracket \,\mathrm{d}\Gamma^0,\\ \int_\Omega \boldsymbol{\sigma} \cdot [\boldsymbol{\nabla} (\delta u)] \,\mathrm{d}\Omega = \int_\Omega \delta u \,\mathrm{d}\Omega + \int_\Gamma \llbracket \delta u\rrbracket \cdot \{\hat{\boldsymbol{\sigma}}\} \,\mathrm{d}\Gamma + \int_{\Gamma^0} \{\delta u\} \llbracket \hat{\boldsymbol{\sigma}}\rrbracket \,\mathrm{d}\Gamma^0,\]

Substituting

\[ \boldsymbol{\tau} = \boldsymbol{\nabla} (\delta u),\\\]

results in

\[ \int_\Omega \boldsymbol{\sigma} \cdot [\boldsymbol{\nabla} (\delta u)] \,\mathrm{d}\Omega = \int_\Omega [\boldsymbol{\nabla} (u)] \cdot [\boldsymbol{\nabla} (\delta u)] \,\mathrm{d}\Omega + \int_\Gamma \llbracket \hat{u} - u\rrbracket \cdot \{\boldsymbol{\nabla} (\delta u)\} \,\mathrm{d}\Gamma + \int_{\Gamma^0} \{\hat{u} - u\} \llbracket \boldsymbol{\nabla} (\delta u)\rrbracket \,\mathrm{d}\Gamma^0,\\ \int_\Omega \boldsymbol{\sigma} \cdot [\boldsymbol{\nabla} (\delta u)] \,\mathrm{d}\Omega = \int_\Omega \delta u \,\mathrm{d}\Omega + \int_\Gamma \llbracket \delta u\rrbracket \cdot \{\hat{\boldsymbol{\sigma}}\} \,\mathrm{d}\Gamma + \int_{\Gamma^0} \{\delta u\} \llbracket \hat{\boldsymbol{\sigma}}\rrbracket \,\mathrm{d}\Gamma^0,\]

Combining the two equations,

\[ \int_\Omega [\boldsymbol{\nabla} (u)] \cdot [\boldsymbol{\nabla} (\delta u)] \,\mathrm{d}\Omega + \int_\Gamma \llbracket \hat{u} - u\rrbracket \cdot \{\boldsymbol{\nabla} (\delta u)\} \,\mathrm{d}\Gamma + \int_{\Gamma^0} \{\hat{u} - u\} \llbracket \boldsymbol{\nabla} (\delta u)\rrbracket \,\mathrm{d}\Gamma^0 - \int_\Gamma \llbracket \delta u\rrbracket \cdot \{\hat{\boldsymbol{\sigma}}\} \,\mathrm{d}\Gamma - \int_{\Gamma^0} \{\delta u\} \llbracket \hat{\boldsymbol{\sigma}}\rrbracket \,\mathrm{d}\Gamma^0 = \int_\Omega \delta u \,\mathrm{d}\Omega,\\\]

The numerical fluxes chosen for the interior penalty method are $\boldsymbol{\hat{\sigma}} = \{\boldsymbol{\nabla} (u)\} - \alpha(\llbracket u\rrbracket)$ on $\Gamma$, $\hat{u} = \{u\}$ on the interfaces between elements $\Gamma^0 : \Gamma \setminus \partial \Omega$, and $\hat{u} = 0$ on $\partial \Omega$. Such choice results in $\{\hat{\boldsymbol{\sigma}}\} = \{\boldsymbol{\nabla} (u)\} - \alpha(\llbracket u\rrbracket)$, $\llbracket \hat{u}\rrbracket = 0$, $\{\hat{u}\} = \{u\}$, $\llbracket \hat{\boldsymbol{\sigma}}\rrbracket = 0$ and the equation becomes

\[ \int_\Omega [\boldsymbol{\nabla} (u)] \cdot [\boldsymbol{\nabla} (\delta u)] \,\mathrm{d}\Omega - \int_\Gamma \llbracket u\rrbracket \cdot \{\boldsymbol{\nabla} (\delta u)\} \,\mathrm{d}\Gamma - \int_\Gamma \llbracket \delta u\rrbracket \cdot \{\boldsymbol{\nabla} (u)\} - \llbracket \delta u\rrbracket \cdot \alpha(\llbracket u\rrbracket) \,\mathrm{d}\Gamma = \int_\Omega \delta u \,\mathrm{d}\Omega,\\\]

Where

\[ \alpha(\llbracket u\rrbracket) = \mu \llbracket u\rrbracket\]

Where $\mu = \eta h_e^{-1}$, the weak form becomes

\[ \int_\Omega [\boldsymbol{\nabla} (u)] \cdot [\boldsymbol{\nabla}] (\delta u) \,\mathrm{d}\Omega - \int_\Gamma \llbracket u \rrbracket \cdot \{\boldsymbol{\nabla} (\delta u)\} + \llbracket \delta u \rrbracket \cdot \{\boldsymbol{\nabla} (u)\} \,\mathrm{d}\Gamma + \int_\Gamma \frac{\eta}{h_e} \llbracket u\rrbracket \cdot \llbracket \delta u\rrbracket \,\mathrm{d}\Gamma = \int_\Omega \delta u \,\mathrm{d}\Omega,\\\]

Since $\partial \Omega$ is constrained with both Dirichlet and Neumann boundary conditions the term $\int_{\partial \Omega} [\boldsymbol{\nabla} (u)] \cdot \boldsymbol{n} \delta u \,\mathrm{d} \Omega$ can be expressed as an integral over $\partial \Omega_N$, where $\partial \Omega_N$ is the boundaries with only prescribed Neumann boundary condition, The resulting weak form is given given as follows: Find $u \in \mathbb{U}$ such that

\[ \int_\Omega [\boldsymbol{\nabla} (u)] \cdot [\boldsymbol{\nabla} (\delta u)] \,\mathrm{d}\Omega - \int_{\Gamma^0} \llbracket u\rrbracket \cdot \{\boldsymbol{\nabla} (\delta u)\} + \llbracket \delta u\rrbracket \cdot \{\boldsymbol{\nabla} (u)\} \,\mathrm{d}\Gamma^0 + \int_{\Gamma^0} \frac{\eta}{h_e} \llbracket u\rrbracket \cdot \llbracket \delta u\rrbracket \,\mathrm{d}\Gamma^0 = \int_\Omega \delta u \,\mathrm{d}\Omega + \int_{\partial \Omega_N} ([\boldsymbol{\nabla} (u)] \cdot \boldsymbol{n}) \delta u \,\mathrm{d} \partial \Omega_N,\\\]

where $h_e$ is the characteristic size (the diameter of the interface), and $\eta$ is a large enough positive number independent of $h_e$ [3], $\delta u \in \mathbb{T}$ is a test function, and where $\mathbb{U}$ and $\mathbb{T}$ are suitable trial and test function sets, respectively. We use the value $\eta = (1 + O)^{D}$, where $O$ is the polynomial order and $D$ the dimension, in this tutorial.

More details on DG formulations for elliptic problems can be found in [4].

Commented Program

Now we solve the problem in Ferrite. What follows is a program spliced with comments. The full program, without comments, can be found in the next section.

First we load Ferrite and other packages, and generate grid just like the heat equation tutorial

using Ferrite, SparseArrays
dim = 2;
grid = generate_grid(Quadrilateral, ntuple(_ -> 20, dim));

We construct the topology information which is used later for generating the sparsity pattern for stiffness matrix.

topology = ExclusiveTopology(grid);

Trial and test functions

CellValues, FacetValues, and InterfaceValues facilitate the process of evaluating values and gradients of test and trial functions (among other things). To define these we need to specify an interpolation space for the shape functions. We use DiscontinuousLagrange functions based on the two-dimensional reference quadrilateral. We also define a quadrature rule based on the same reference element. We combine the interpolation and the quadrature rule to CellValues and InterfaceValues object. Note that InterfaceValues object contains two FacetValues objects which can be used individually.

order = 1;
ip = DiscontinuousLagrange{RefQuadrilateral, order}();
qr = QuadratureRule{RefQuadrilateral}(2);

For FacetValues and InterfaceValues we use FacetQuadratureRule

facet_qr = FacetQuadratureRule{RefQuadrilateral}(2);
cellvalues = CellValues(qr, ip);
facetvalues = FacetValues(facet_qr, ip);
interfacevalues = InterfaceValues(facet_qr, ip);

Penalty term parameters

We define functions to calculate the diameter of a set of points, used to calculate the characteristic size $h_e$ in the assembly routine.

getdistance(p1::Vec{N, T},p2::Vec{N, T}) where {N, T} = norm(p1-p2);
getdiameter(cell_coords::Vector{Vec{N, T}}) where {N, T} = maximum(getdistance.(cell_coords, reshape(cell_coords, (1,:))));

Degrees of freedom

Degrees of freedom distribution is handled using DofHandler as usual

dh = DofHandler(grid)
add!(dh, :u, ip)
close!(dh);

However, when generating the sparsity pattern we need to pass the topology and the cross-element coupling matrix when we're using discontinuous interpolations. The cross-element coupling matrix is of size [1,1] in this case as we have only one field and one DofHandler.

K = allocate_matrix(dh, topology = topology, interface_coupling = trues(1,1));

Boundary conditions

The Dirichlet boundary conditions are treated as usual by a ConstraintHandler.

ch = ConstraintHandler(dh)
add!(ch, Dirichlet(:u, getfacetset(grid, "right"), (x, t) -> 1.0))
add!(ch, Dirichlet(:u, getfacetset(grid, "left"), (x, t) -> -1.0))
close!(ch);

Furthermore, we define $\partial \Omega_N$ as the union of the facet sets with Neumann boundary conditions for later use

∂Ωₙ = union(
    getfacetset(grid, "top"),
    getfacetset(grid, "bottom"),
);

Assembling the linear system

Now we have all the pieces needed to assemble the linear system, $K u = f$. Assembling of the global system is done by looping over i) all the elements in order to compute the element contributions $K_e$ and $f_e$, ii) all the interfaces to compute their contributions $K_i$, and iii) all the Neumann boundary facets to compute their contributions $f_e$. All these local contributions are then assembled into the appropriate place in the global $K$ and $f$.

Local assembly

We define the functions

assemble_element! to compute the contributions $K_e$ and $f_e$ of volume integrals over an element using cellvalues.
assemble_interface! to compute the contribution $K_i$ of surface integrals over an interface using interfacevalues.
assemble_boundary! to compute the contribution $f_e$ of surface integrals over a boundary facet using FacetValues.

function assemble_element!(Ke::Matrix, fe::Vector, cellvalues::CellValues)
    n_basefuncs = getnbasefunctions(cellvalues)
    # Reset to 0
    fill!(Ke, 0)
    fill!(fe, 0)
    # Loop over quadrature points
    for q_point in 1:getnquadpoints(cellvalues)
        # Quadrature weight
        dΩ = getdetJdV(cellvalues, q_point)
        # Loop over test shape functions
        for i in 1:n_basefuncs
            δu  = shape_value(cellvalues, q_point, i)
            ∇δu = shape_gradient(cellvalues, q_point, i)
            # Add contribution to fe
            fe[i] += δu * dΩ
            # Loop over trial shape functions
            for j in 1:n_basefuncs
                ∇u = shape_gradient(cellvalues, q_point, j)
                # Add contribution to Ke
                Ke[i, j] += (∇δu ⋅ ∇u) * dΩ
            end
        end
    end
    return Ke, fe
end

function assemble_interface!(Ki::Matrix, iv::InterfaceValues, μ::Float64)
    # Reset to 0
    fill!(Ki, 0)
    # Loop over quadrature points
    for q_point in 1:getnquadpoints(iv)
        # Get the normal to facet A
        normal = getnormal(iv, q_point)
        # Get the quadrature weight
        dΓ = getdetJdV(iv, q_point)
        # Loop over test shape functions
        for i in 1:getnbasefunctions(iv)
            # Multiply the jump by the negative normal to get the definition from the theory section.
            δu_jump = shape_value_jump(iv, q_point, i) * (-normal)
            ∇δu_avg = shape_gradient_average(iv, q_point, i)
            # Loop over trial shape functions
            for j in 1:getnbasefunctions(iv)
                # Multiply the jump by the negative normal to get the definition from the theory section.
                u_jump = shape_value_jump(iv, q_point, j) * (-normal)
                ∇u_avg = shape_gradient_average(iv, q_point, j)
                # Add contribution to Ki
                Ki[i, j] += -(δu_jump ⋅ ∇u_avg + ∇δu_avg ⋅ u_jump)*dΓ +  μ * (δu_jump ⋅ u_jump) * dΓ
            end
        end
    end
    return Ki
end

function assemble_boundary!(fe::Vector, fv::FacetValues)
    # Reset to 0
    fill!(fe, 0)
    # Loop over quadrature points
    for q_point in 1:getnquadpoints(fv)
        # Get the normal to facet A
        normal = getnormal(fv, q_point)
        # Get the quadrature weight
        ∂Ω = getdetJdV(fv, q_point)
        # Loop over test shape functions
        for i in 1:getnbasefunctions(fv)
            δu = shape_value(fv, q_point, i)
            boundary_flux = normal[2]
            fe[i] = boundary_flux * δu * ∂Ω
        end
    end
    return fe
end

Global assembly

We define the function assemble_global to loop over all elements and internal facets (interfaces), as well as the external facets involved in Neumann boundary conditions.

function assemble_global(cellvalues::CellValues, facetvalues::FacetValues, interfacevalues::InterfaceValues, K::SparseMatrixCSC, dh::DofHandler, order::Int, dim::Int)
    # Allocate the element stiffness matrix and element force vector
    n_basefuncs = getnbasefunctions(cellvalues)
    Ke = zeros(n_basefuncs, n_basefuncs)
    fe = zeros(n_basefuncs)
    Ki = zeros(n_basefuncs * 2, n_basefuncs * 2)
    # Allocate global force vector f
    f = zeros(ndofs(dh))
    # Create an assembler
    assembler = start_assemble(K, f)
    # Loop over all cells
    for cell in CellIterator(dh)
        # Reinitialize cellvalues for this cell
        reinit!(cellvalues, cell)
        # Compute volume integral contribution
        assemble_element!(Ke, fe, cellvalues)
        # Assemble Ke and fe into K and f
        assemble!(assembler, celldofs(cell), Ke, fe)
    end
    # Loop over all interfaces
    for ic in InterfaceIterator(dh)
        # Reinitialize interfacevalues for this interface
        reinit!(interfacevalues, ic)
        # Calculate the characteristic size hₑ as the face diameter
        interfacecoords =  ∩(getcoordinates(ic)...)
        hₑ = getdiameter(interfacecoords)
        # Calculate μ
        μ = (1 + order)^dim / hₑ
        # Compute interface surface integrals contribution
        assemble_interface!(Ki, interfacevalues, μ)
        # Assemble Ki into K
        assemble!(assembler, interfacedofs(ic), Ki)
    end
    # Loop over domain boundaries with Neumann boundary conditions
    for fc in FacetIterator(dh, ∂Ωₙ)
        # Reinitialize facetvalues for this boundary facet
        reinit!(facetvalues, fc)
        # Compute boundary facet surface integrals contribution
        assemble_boundary!(fe, facetvalues)
        # Assemble fe into f
        assemble!(f, celldofs(fc), fe)
    end
    return K, f
end
K, f = assemble_global(cellvalues, facetvalues, interfacevalues, K, dh, order, dim);

Solution of the system

The solution of the system is independent of the discontinuous discretization and the application of constraints, linear solve, and exporting is done as usual.

apply!(K, f, ch)
u = K \ f;
VTKGridFile("dg_heat_equation", dh) do vtk
    write_solution(vtk, dh, u)
end;

References

[3]: L. Mu, J. Wang, Y. Wang and X. Ye. Interior penalty discontinuous Galerkin method on very general polygonal and polyhedral meshes. Journal of Computational and Applied Mathematics 255, 432–440 (2014).
[4]: D. N. Arnold, F. Brezzi, B. Cockburn and L. D. Marini. Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems. SIAM Journal on Numerical Analysis 39, 1749–1779 (2002). Accessed on Dec 20, 2023.

Plain program

Here follows a version of the program without any comments. The file is also available here: dg_heat_equation.jl.

using Ferrite, SparseArrays
dim = 2;
grid = generate_grid(Quadrilateral, ntuple(_ -> 20, dim));

topology = ExclusiveTopology(grid);

order = 1;
ip = DiscontinuousLagrange{RefQuadrilateral, order}();
qr = QuadratureRule{RefQuadrilateral}(2);

facet_qr = FacetQuadratureRule{RefQuadrilateral}(2);
cellvalues = CellValues(qr, ip);
facetvalues = FacetValues(facet_qr, ip);
interfacevalues = InterfaceValues(facet_qr, ip);

getdistance(p1::Vec{N, T},p2::Vec{N, T}) where {N, T} = norm(p1-p2);
getdiameter(cell_coords::Vector{Vec{N, T}}) where {N, T} = maximum(getdistance.(cell_coords, reshape(cell_coords, (1,:))));

dh = DofHandler(grid)
add!(dh, :u, ip)
close!(dh);

K = allocate_matrix(dh, topology = topology, interface_coupling = trues(1,1));

ch = ConstraintHandler(dh)
add!(ch, Dirichlet(:u, getfacetset(grid, "right"), (x, t) -> 1.0))
add!(ch, Dirichlet(:u, getfacetset(grid, "left"), (x, t) -> -1.0))
close!(ch);

∂Ωₙ = union(
    getfacetset(grid, "top"),
    getfacetset(grid, "bottom"),
);

function assemble_element!(Ke::Matrix, fe::Vector, cellvalues::CellValues)
    n_basefuncs = getnbasefunctions(cellvalues)
    # Reset to 0
    fill!(Ke, 0)
    fill!(fe, 0)
    # Loop over quadrature points
    for q_point in 1:getnquadpoints(cellvalues)
        # Quadrature weight
        dΩ = getdetJdV(cellvalues, q_point)
        # Loop over test shape functions
        for i in 1:n_basefuncs
            δu  = shape_value(cellvalues, q_point, i)
            ∇δu = shape_gradient(cellvalues, q_point, i)
            # Add contribution to fe
            fe[i] += δu * dΩ
            # Loop over trial shape functions
            for j in 1:n_basefuncs
                ∇u = shape_gradient(cellvalues, q_point, j)
                # Add contribution to Ke
                Ke[i, j] += (∇δu ⋅ ∇u) * dΩ
            end
        end
    end
    return Ke, fe
end

function assemble_interface!(Ki::Matrix, iv::InterfaceValues, μ::Float64)
    # Reset to 0
    fill!(Ki, 0)
    # Loop over quadrature points
    for q_point in 1:getnquadpoints(iv)
        # Get the normal to facet A
        normal = getnormal(iv, q_point)
        # Get the quadrature weight
        dΓ = getdetJdV(iv, q_point)
        # Loop over test shape functions
        for i in 1:getnbasefunctions(iv)
            # Multiply the jump by the negative normal to get the definition from the theory section.
            δu_jump = shape_value_jump(iv, q_point, i) * (-normal)
            ∇δu_avg = shape_gradient_average(iv, q_point, i)
            # Loop over trial shape functions
            for j in 1:getnbasefunctions(iv)
                # Multiply the jump by the negative normal to get the definition from the theory section.
                u_jump = shape_value_jump(iv, q_point, j) * (-normal)
                ∇u_avg = shape_gradient_average(iv, q_point, j)
                # Add contribution to Ki
                Ki[i, j] += -(δu_jump ⋅ ∇u_avg + ∇δu_avg ⋅ u_jump)*dΓ +  μ * (δu_jump ⋅ u_jump) * dΓ
            end
        end
    end
    return Ki
end

function assemble_boundary!(fe::Vector, fv::FacetValues)
    # Reset to 0
    fill!(fe, 0)
    # Loop over quadrature points
    for q_point in 1:getnquadpoints(fv)
        # Get the normal to facet A
        normal = getnormal(fv, q_point)
        # Get the quadrature weight
        ∂Ω = getdetJdV(fv, q_point)
        # Loop over test shape functions
        for i in 1:getnbasefunctions(fv)
            δu = shape_value(fv, q_point, i)
            boundary_flux = normal[2]
            fe[i] = boundary_flux * δu * ∂Ω
        end
    end
    return fe
end

function assemble_global(cellvalues::CellValues, facetvalues::FacetValues, interfacevalues::InterfaceValues, K::SparseMatrixCSC, dh::DofHandler, order::Int, dim::Int)
    # Allocate the element stiffness matrix and element force vector
    n_basefuncs = getnbasefunctions(cellvalues)
    Ke = zeros(n_basefuncs, n_basefuncs)
    fe = zeros(n_basefuncs)
    Ki = zeros(n_basefuncs * 2, n_basefuncs * 2)
    # Allocate global force vector f
    f = zeros(ndofs(dh))
    # Create an assembler
    assembler = start_assemble(K, f)
    # Loop over all cells
    for cell in CellIterator(dh)
        # Reinitialize cellvalues for this cell
        reinit!(cellvalues, cell)
        # Compute volume integral contribution
        assemble_element!(Ke, fe, cellvalues)
        # Assemble Ke and fe into K and f
        assemble!(assembler, celldofs(cell), Ke, fe)
    end
    # Loop over all interfaces
    for ic in InterfaceIterator(dh)
        # Reinitialize interfacevalues for this interface
        reinit!(interfacevalues, ic)
        # Calculate the characteristic size hₑ as the face diameter
        interfacecoords =  ∩(getcoordinates(ic)...)
        hₑ = getdiameter(interfacecoords)
        # Calculate μ
        μ = (1 + order)^dim / hₑ
        # Compute interface surface integrals contribution
        assemble_interface!(Ki, interfacevalues, μ)
        # Assemble Ki into K
        assemble!(assembler, interfacedofs(ic), Ki)
    end
    # Loop over domain boundaries with Neumann boundary conditions
    for fc in FacetIterator(dh, ∂Ωₙ)
        # Reinitialize facetvalues for this boundary facet
        reinit!(facetvalues, fc)
        # Compute boundary facet surface integrals contribution
        assemble_boundary!(fe, facetvalues)
        # Assemble fe into f
        assemble!(f, celldofs(fc), fe)
    end
    return K, f
end
K, f = assemble_global(cellvalues, facetvalues, interfacevalues, K, dh, order, dim);

apply!(K, f, ch)
u = K \ f;
VTKGridFile("dg_heat_equation", dh) do vtk
    write_solution(vtk, dh, u)
end;

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