<br>

<i>This program was contributed by Luca Heltai (thanks to Michael
Gratton for pointing out what the exact solution should have been in
the three dimensional case).  </i>

@dealiiTutorialDOI{10.5281/zenodo.495473,https://zenodo.org/badge/DOI/10.5281/zenodo.495473.svg}

<a name="step-34-Intro"></a>

<h1>Introduction</h1>

<h3> Irrotational flow </h3>
The incompressible motion of an inviscid fluid past a body (for
example air past an airplane wing, or air or water past a propeller) is
usually modeled by the Euler equations of fluid dynamics:

\f{align*}
  \frac{\partial }{\partial t}\mathbf{v} + (\mathbf{v}\cdot\nabla)\mathbf{v}
  &=
  -\frac{1}{\rho}\nabla p + \mathbf{g}
  \qquad &\text{in } \mathbb{R}^n \backslash \Omega
  \\
  \nabla \cdot \mathbf{v}&=0
  &\text{in } \mathbb{R}^n\backslash\Omega
\f}
where the fluid density $\rho$ and the acceleration $\mathbf{g}$ due
to external forces are given and the velocity $\mathbf{v}$ and the
pressure $p$ are the unknowns. Here $\Omega$ is a closed bounded
region representing the body around which the fluid moves.

The above equations can be derived from Navier-Stokes equations
assuming that the effects due to viscosity are negligible compared to
those due to the pressure gradient, inertial forces and the external
forces. This is the opposite case of the Stokes equations discussed in
step-22 which are the limit case of dominant viscosity,
i.e. where the velocity is so small that inertia forces can be
neglected. On the other hand, owing to the assumed incompressibility,
the equations are not suited for very high speed gas flows where
compressibility and the equation of state of the gas have to be taken
into account, leading to the Euler equations of gas dynamics, a
hyperbolic system.

For the purpose of this tutorial program, we will consider only stationary
flow without external forces:
\f{align*}
  (\mathbf{v}\cdot\nabla)\mathbf{v}
  &=
  -\frac{1}{\rho}\nabla p
  \qquad &\text{in } \mathbb{R}^n \backslash \Omega
  \\
  \nabla \cdot \mathbf{v}&=0
  &\text{in } \mathbb{R}^n\backslash\Omega
\f}


Uniqueness of the solution of the Euler equations is ensured by adding the
boundary conditions
\f[
  \label{eq:boundary-conditions}
  \begin{aligned}
    \mathbf{n}\cdot\mathbf{v}& = 0 \qquad && \text{ on } \partial\Omega \\
    \mathbf{v}& = \mathbf{v}_\infty && \text{ when } |\mathbf{x}| \to \infty,
  \end{aligned}
\f]

which is to say that the body is at rest in our coordinate systems and
is not permeable, and that the fluid has (constant) velocity
$\mathbf{v}_\infty$ at infinity. An alternative viewpoint is that our
coordinate system moves along with the body whereas the background
fluid is at rest at infinity. Notice that we define the normal
$\mathbf{n}$ as the <i>outer</i> normal to the domain $\Omega$, which
is the opposite of the outer normal to the integration domain.

For both stationary and non stationary flow, the solution process
starts by solving for the velocity in the second equation and
substituting in the first equation in order to find the pressure.
The solution of the stationary Euler equations is typically performed
in order to understand the behavior of the given (possibly complex)
geometry when a prescribed motion is enforced on the system.

The first step in this process is to change the frame of reference from a
coordinate system moving along with the body to one in which the body moves
through a fluid that is at rest at infinity. This can be expressed by
introducing a new velocity $\mathbf{\tilde{v}}=\mathbf{v}-\mathbf{v}_\infty$ for
which we find that the same equations hold (because $\nabla\cdot
\mathbf{v}_\infty=0$) and we have boundary conditions
\f[
  \label{eq:boundary-conditions-tilde}
  \begin{aligned}
    \mathbf{n}\cdot\mathbf{\tilde{v}}& = -\mathbf{n}\cdot\mathbf{v}_\infty \qquad && \text{ on } \partial\Omega \\
    \mathbf{\tilde{v}}& = 0 && \text{ when } |\mathbf{x}| \to \infty,
  \end{aligned}
\f]

If we assume that the fluid is irrotational, i.e., $\nabla \times
\mathbf{v}=0$ in $\mathbb{R}^n\backslash\Omega$, we can represent the
velocity, and consequently also the perturbation velocity, as the
gradient of a scalar function:
\f[
  \mathbf{\tilde{v}}=\nabla\phi,
\f]
and so the second part of Euler equations above can be rewritten
as the homogeneous Laplace equation for the unknown $\phi$:
\f{align*}
\label{laplace}
\Delta\phi &= 0 \qquad &&\text{in}\ \mathbb{R}^n\backslash\Omega,
	   \\
	   \mathbf{n}\cdot\nabla\phi &= -\mathbf{n}\cdot\mathbf{v}_\infty
	   && \text{on}\ \partial\Omega
\f}
while the momentum equation reduces to Bernoulli's equation that expresses the
pressure $p$ as a function of the potential $\phi$:
\f[
\frac{p}{\rho} +\frac{1}{2} | \nabla \phi |^2 = 0 \in \Omega.
\f]

So we can solve the problem by solving the Laplace equation for the
potential.  We recall that the following functions, called fundamental
solutions of the Laplace equation,

\f[ \begin{aligned}
\label{eq:3} G(\mathbf{y}-\mathbf{x}) = &
-\frac{1}{2\pi}\ln|\mathbf{y}-\mathbf{x}| \qquad && \text{for } n=2 \\
G(\mathbf{y}-\mathbf{x}) = &
\frac{1}{4\pi}\frac{1}{|\mathbf{y}-\mathbf{x}|}&& \text{for } n=3,
\end{aligned}
\f]

satisfy in a distributional sense the equation:

\f[
-\Delta_y G(\mathbf{y}-\mathbf{x}) = \delta(\mathbf{y}-\mathbf{x}),
\f]

where the derivative is done in the variable $\mathbf{y}$. By using
the usual Green identities, our problem can be written on the boundary
$\partial\Omega = \Gamma$ only. We recall the general definition of
the second Green %identity:

\f[\label{green}
  \int_{\omega}
  (-\Delta u)v\,dx + \int_{\partial\omega} \frac{\partial u}{\partial \tilde{\mathbf{n}} }v \,ds
  =
  \int_{\omega}
  (-\Delta v)u\,dx + \int_{\partial\omega} u\frac{\partial v}{\partial \tilde{\mathbf{n}}} \,ds,
\f]

where $\tilde{\mathbf{n}}$ is the normal to the surface of $\omega$ pointing
outwards from the domain of integration $\omega$.

In our case the domain of integration is the domain
$\mathbb{R}^n\backslash\Omega$, whose boundary is $ \Gamma_\infty \cup
\Gamma$, where the "boundary" at infinity is defined as

\f[
\Gamma_\infty \dealcoloneq \lim_{r\to\infty} \partial B_r(0).
\f]

In our program the normals are defined as <i>outer</i> to the domain
$\Omega$, that is, they are in fact <i>inner</i> to the integration
domain, and some care is required in defining the various integrals
with the correct signs for the normals, i.e. replacing $\tilde{\mathbf{n}}$
by $-\mathbf{n}$.

If we substitute $u$ and $v$ in the Green %identity with the solution
$\phi$ and with the fundamental solution of the Laplace equation
respectively, as long as $\mathbf{x}$ is chosen in the region
$\mathbb{R}^n\backslash\Omega$, we obtain:
\f[
  \phi(\mathbf{x}) -
  \int_{\Gamma\cup\Gamma_\infty}\frac{\partial G(\mathbf{y}-\mathbf{x})}{\partial \mathbf{n}_y}\phi(\mathbf{y})\,ds_y
  =
  -\int_{\Gamma\cup\Gamma_\infty}G(\mathbf{y}-\mathbf{x})\frac{\partial \phi}{\partial \mathbf{n}_y}(\mathbf{y})\,ds_y
  \qquad \forall\mathbf{x}\in \mathbb{R}^n\backslash\Omega
\f]

where the normals are now pointing <i>inward</i> the domain of
integration.

Notice that in the above equation, we also have the integrals on the
portion of the boundary at $\Gamma_\infty$. Using the boundary
conditions of our problem, we have that $\nabla \phi$ is zero at
infinity (which simplifies the integral on $\Gamma_\infty$ on the
right hand side).

The integral on $\Gamma_\infty$ that appears on the left hand side can
be treated by observing that $\nabla\phi=0$ implies that $\phi$ at
infinity is necessarily constant. We define its value to be
$\phi_\infty$.  It is an easy exercise to prove that

\f[
-\int_{\Gamma_\infty} \frac{\partial G(\mathbf{y}-\mathbf{x})}
{\partial \mathbf{n}_y}\phi_\infty \,ds_y =
\lim_{r\to\infty} \int_{\partial B_r(0)} \frac{\mathbf{r}}{r} \cdot \nabla G(\mathbf{y}-\mathbf{x})
\phi_\infty \,ds_y = -\phi_\infty.
\f]

Using this result, we can reduce the above equation only on the
boundary $\Gamma$ using the so-called Single and Double Layer
Potential operators:

\f[\label{integral}
  \phi(\mathbf{x}) - (D\phi)(\mathbf{x}) = \phi_\infty
  -\left(S \frac{\partial \phi}{\partial n_y}\right)(\mathbf{x})
  \qquad \forall\mathbf{x}\in \mathbb{R}^n\backslash\Omega.
\f]

(The name of these operators comes from the fact that they describe the
electric potential in $\mathbb{R}^n$ due to a single thin sheet of charges
along a surface, and due to a double sheet of charges and anti-charges along
the surface, respectively.)

In our case, we know the Neumann values of $\phi$ on the boundary:
$\mathbf{n}\cdot\nabla\phi = -\mathbf{n}\cdot\mathbf{v}_\infty$.
Consequently,
\f[
  \phi(\mathbf{x}) - (D\phi)(\mathbf{x}) = \phi_\infty +
   \left(S[\mathbf{n}\cdot\mathbf{v}_\infty]\right)(\mathbf{x})
   \qquad \forall\mathbf{x} \in \mathbb{R}^n\backslash\Omega.
\f]
If we take the limit for $\mathbf{x}$ tending to $\Gamma$ of
the above equation, using well known properties of the single and double layer
operators, we obtain an equation for $\phi$ just on the boundary $\Gamma$ of
$\Omega$:

\f[\label{SD}
  \alpha(\mathbf{x})\phi(\mathbf{x}) - (D\phi)(\mathbf{x}) = \phi_\infty +
  \left(S [\mathbf{n}\cdot\mathbf{v}_\infty]\right)(\mathbf{x})
  \quad \mathbf{x}\in \partial\Omega,
\f]

which is the Boundary Integral Equation (BIE) we were looking for,
where the quantity $\alpha(\mathbf{x})$ is the fraction of angle or
solid angle by which the point $\mathbf{x}$ sees the domain of
integration $\mathbb{R}^n\backslash\Omega$.

In particular, at points $\mathbf{x}$ where the boundary
$\partial\Omega$ is differentiable (i.e. smooth) we have
$\alpha(\mathbf{x})=\frac 12$, but the value may be smaller or larger
at points where the boundary has a corner or an edge.

Substituting the single and double layer operators we get:
\f[
  \alpha(\mathbf{x}) \phi(\mathbf{x})
  + \frac{1}{2\pi}\int_{\partial \Omega}  \frac{
  (\mathbf{y}-\mathbf{x})\cdot\mathbf{n}_y  }{ |\mathbf{y}-\mathbf{x}|^2 }
  \phi(\mathbf{y}) \,ds_y
  = \phi_\infty
    -\frac{1}{2\pi}\int_{\partial \Omega}  \ln|\mathbf{y}-\mathbf{x}| \, \mathbf{n}\cdot\mathbf{v_\infty}\,ds_y
\f]
for two dimensional flows and
\f[
  \alpha(\mathbf{x}) \phi(\mathbf{x})
   + \frac{1}{4\pi}\int_{\partial \Omega} \frac{ (\mathbf{y}-\mathbf{x})\cdot\mathbf{n}_y  }{ |\mathbf{y}-\mathbf{x}|^3 }\phi(\mathbf{y})\,ds_y
  = \phi_\infty +
  \frac{1}{4\pi}\int_{\partial \Omega} \frac{1}{|\mathbf{y}-\mathbf{x}|} \, \mathbf{n}\cdot\mathbf{v_\infty}\,ds_y
\f]
for three dimensional flows, where the normal derivatives of the fundamental
solutions have been written in a form that makes computation easier. In either
case, $\phi$ is the solution of an integral equation posed entirely on the
boundary since both $\mathbf{x},\mathbf{y}\in\partial\Omega$.

Notice that the fraction of angle (in 2d) or solid angle (in 3d)
$\alpha(\mathbf{x})$ by which the point $\mathbf{x}$ sees the domain
$\Omega$ can be defined using the double layer potential itself:
\f[
\alpha(\mathbf{x}) \dealcoloneq 1 -
\frac{1}{2(n-1)\pi}\int_{\partial \Omega} \frac{ (\mathbf{y}-\mathbf{x})\cdot\mathbf{n}_y  }
{ |\mathbf{y}-\mathbf{x}|^{n} }\phi(\mathbf{y})\,ds_y = 1+
\int_{\partial \Omega} \frac{ \partial G(\mathbf{y}-\mathbf{x}) }{\partial \mathbf{n}_y} \, ds_y.
\f]

The reason why this is possible can be understood if we consider the
fact that the solution of a pure Neumann problem is known up to an
arbitrary constant $c$, which means that, if we set the Neumann data
to be zero, then any constant $\phi = \phi_\infty$ will be a solution.
Inserting the constant solution and the Neumann boundary condition in the
boundary integral equation, we have
@f{align*}{
\alpha\left(\mathbf{x}\right)\phi\left(\mathbf{x}\right)
&=\int_{\Omega}\phi\left(\mathbf{y}\right)\delta\left(\mathbf{y}-\mathbf{x}\right)\, dy\\
\Rightarrow
\alpha\left(\mathbf{x}\right)\phi_\infty
&=\phi_\infty\int_{\Gamma\cup\Gamma_\infty}\frac{ \partial G(\mathbf{y}-\mathbf{x}) }{\partial \mathbf{n}_y} \, ds_y
=\phi_\infty\left[\int_{\Gamma_\infty}\frac{ \partial G(\mathbf{y}-\mathbf{x}) }{\partial \mathbf{n}_y} \, ds_y
+\int_{\Gamma}\frac{ \partial G(\mathbf{y}-\mathbf{x}) }{\partial \mathbf{n}_y} \, ds_y
\right]
@f}
The integral on $\Gamma_\infty$ is unity, see above, so division by the constant $\phi_\infty$ gives us the explicit
expression above for $\alpha(\mathbf{x})$.

While this example program is really only focused on the solution of the
boundary integral equation, in a realistic setup one would still need to solve
for the velocities. To this end, note that we have just computed
$\phi(\mathbf{x})$ for all $\mathbf{x}\in\partial\Omega$. In the next step, we
can compute (analytically, if we want) the solution $\phi(\mathbf{x})$ in all
of $\mathbb{R}^n\backslash\Omega$. To this end, recall that we had
\f[
  \phi(\mathbf{x})
  =
  \phi_\infty +
  (D\phi)(\mathbf{x})
  +
  \left(S[\mathbf{n}\cdot\mathbf{v}_\infty]\right)(\mathbf{x})
  \qquad \forall\mathbf{x}\in \mathbb{R}^n\backslash\Omega.
\f]
where now we have everything that is on the right hand side ($S$ and $D$ are
integrals we can evaluate, the normal velocity on the boundary is given, and
$\phi$ on the boundary we have just computed). Finally, we can then recover
the velocity as $\mathbf{\tilde v}=\nabla \phi$.

Notice that the evaluation of the above formula for $\mathbf{x} \in
\Omega$ should yield zero as a result, since the integration of the
Dirac delta $\delta(\mathbf{x})$ in the domain
$\mathbb{R}^n\backslash\Omega$ is always zero by definition.

As a final test, let us verify that this velocity indeed satisfies the
momentum balance equation for a stationary flow field, i.e., whether
$\mathbf{v}\cdot\nabla\mathbf{v} = -\frac 1\rho \nabla p$ where
$\mathbf{v}=\mathbf{\tilde
v}+\mathbf{v}_\infty=\nabla\phi+\mathbf{v}_\infty$ for some (unknown) pressure
$p$ and a given constant $\rho$. In other words, we would like to verify that
Bernoulli's law as stated above indeed holds. To show this, we use that
the left hand side of this equation equates to
@f{align*}{
  \mathbf{v}\cdot\nabla\mathbf{v}
  &=
  [(\nabla\phi+\mathbf{v}_\infty)\cdot\nabla] (\nabla\phi+\mathbf{v}_\infty)
  \\
  &=
  [(\nabla\phi+\mathbf{v}_\infty)\cdot\nabla] (\nabla\phi)
@f}
where we have used that $\mathbf{v}_\infty$ is constant. We would like to
write this expression as the gradient of something (remember that $\rho$ is a
constant). The next step is more
convenient if we consider the components of the equation individually
(summation over indices that appear twice is implied):
@f{align*}{
  [\mathbf{v}\cdot\nabla\mathbf{v}]_i
  &=
  (\partial_j\phi+v_{\infty,j}) \partial_j \partial_i\phi
  \\
  &=
  \partial_j [(\partial_j\phi+v_{\infty,j}) \partial_i\phi]
  -
  \partial_j [(\partial_j\phi+v_{\infty,j})] \partial_i\phi
  \\
  &=
  \partial_j [(\partial_j\phi+v_{\infty,j}) \partial_i\phi]
@f}
because $\partial_j \partial_j\phi = \Delta \phi = 0$ and $\textrm{div}
\ \mathbf{v}_\infty=0$. Next,
@f{align*}{
  [\mathbf{v}\cdot\nabla\mathbf{v}]_i
  &=
  \partial_j [(\partial_j\phi+v_{\infty,j}) \partial_i\phi]
  \\
  &=
  \partial_j [(\partial_j\phi) (\partial_i\phi)]
  +
  \partial_j [v_{\infty,j} \partial_i\phi]
  \\
  &=
  \partial_j [(\partial_j\phi) (\partial_i\phi)]
  +
  \partial_j [v_{\infty,j}] \partial_i\phi
  +
  v_{\infty,j} \partial_j \partial_i\phi
  \\
  &=
  \partial_j [(\partial_j\phi) (\partial_i\phi)]
  +
  v_{\infty,j} \partial_j \partial_i\phi
  \\
  &=
  \partial_i \partial_j [(\partial_j\phi) \phi]
  -
  \partial_j [\partial_i (\partial_j\phi) \phi]
  +
  \partial_i [v_{\infty,j} \partial_j \phi]
  -
  \partial_i [v_{\infty,j}] \partial_j \phi
@f}
Again, the last term disappears because $\mathbf{v}_\infty$ is constant and we
can merge the first and third term into one:
@f{align*}{
  [\mathbf{v}\cdot\nabla\mathbf{v}]_i
  &=
  \partial_i (\partial_j [(\partial_j\phi) \phi + v_{\infty,j} \partial_j \phi])
  -
  \partial_j [\partial_i (\partial_j\phi) \phi]
  \\
  &=
  \partial_i [(\partial_j\phi)(\partial_j \phi) + v_{\infty,j} \partial_j \phi]
  -
  \partial_j [\partial_i (\partial_j\phi) \phi]
@f}

We now only need to massage that last term a bit more. Using the product rule,
we get
@f{align*}{
  \partial_j [\partial_i (\partial_j\phi) \phi]
  &=
  \partial_i [\partial_j \partial_j\phi] \phi
  +
  \partial_i [\partial_j \phi] (\partial_j \phi).
@f}
The first of these terms is zero (because, again, the summation over $j$ gives
$\Delta\phi$, which is zero). The last term can be written as $\frac 12
\partial_i [(\partial_j\phi)(\partial_j\phi)]$ which is in the desired gradient
form. As a consequence, we can now finally state that
@f{align*}{
  [\mathbf{v}\cdot\nabla\mathbf{v}]_i
  &=
  \partial_i (\partial_j [(\partial_j\phi) \phi + v_{\infty,j} \partial_j \phi])
  -
  \partial_j [\partial_i (\partial_j\phi) \phi]
  \\
  &=
  \partial_i
  \left[
    (\partial_j\phi)(\partial_j \phi) + v_{\infty,j} \partial_j \phi
    -
    \frac 12 (\partial_j\phi)(\partial_j\phi)
  \right],
  \\
  &=
  \partial_i
  \left[
    \frac 12 (\partial_j\phi)(\partial_j \phi) + v_{\infty,j} \partial_j \phi
  \right],
@f}
or in vector form:
@f[
  \mathbf{v}\cdot\nabla\mathbf{v}
  =
  \nabla
  \left[
    \frac 12 \mathbf{\tilde v}^2
    + \mathbf{v}_{\infty} \cdot \mathbf{\tilde v}
  \right],
@f]
or in other words:
@f[
  p
  =
  -\rho
  \left[
    \frac 12 \mathbf{\tilde v}^2
    + \mathbf{v}_{\infty} \cdot \mathbf{\tilde v}
  \right]
  =
  -\rho
  \left[
    \frac 12 \mathbf{v}^2
    -
    \frac 12 \mathbf{v}_{\infty}^2
  \right]
  .
@f]
Because the pressure is only determined up to a constant (it appears only with
a gradient in the equations), an equally valid definition is
@f[
  p
  =
  -\frac 12 \rho \mathbf{v}^2
  .
@f]
This is exactly Bernoulli's law mentioned above.


<h3>The numerical approximation</h3>

Numerical approximations of Boundary Integral Equations (BIE) are commonly
referred to as the boundary element method or panel method (the latter
expression being used mostly in the computational fluid dynamics community).
The goal of the following test problem is to solve the integral
formulation of the Laplace equation with Neumann boundary conditions,
using a circle and a sphere respectively in two and three space
dimensions, illustrating along the way the features that allow one to
treat boundary element problems almost as easily as finite element
problems using the deal.II library.

To this end, let $\mathcal{T}_h = \bigcup_i K_i$ be a subdivision of the
manifold $\Gamma = \partial \Omega$ into $M$ line segments if $n=2$, or $M$
quadrilaterals if $n=3$. We will call each individual segment or
quadrilateral an <i>element</i> or <i>cell</i>, independently of the
dimension $n$ of the surrounding space $\mathbb{R}^n$.
We define the finite dimensional space $V_h$ as
\f[
  \label{eq:definition-Vh}
  V_h \dealcoloneq \{ v \in C^0(\Gamma) \text{ s.t. } v|_{K_i} \in \mathcal{Q}^1(K_i),
  \forall i\},
\f]
with basis functions $\psi_i(\mathbf{x})$ for which we will use the usual FE_Q
finite element, with the catch that this time it is defined on a manifold of
codimension one (which we do by using the second template argument that is
usually defaulted to equal the first; here, we will create objects
<code>FE_Q@<dim-1,dim@></code> to indicate that we have <code>dim-1</code>
dimensional cells in a <code>dim</code> dimensional space).
An element $\phi_h$ of $V_h$ is uniquely
identified by the vector $\boldsymbol{\phi}$ of its coefficients
$\phi_i$, that is:
\f[
  \label{eq:definition-of-element}
  \phi_h(\mathbf{x}) \dealcoloneq \phi_i \psi_i(\mathbf{x}), \qquad
  \boldsymbol{\phi} \dealcoloneq \{ \phi_i \},
\f]
where summation  is implied over repeated indexes. Note that we could use
discontinuous elements here &mdash; in fact, there is no real reason to use
continuous ones since the integral formulation does not
imply any derivatives on our trial functions so continuity is unnecessary,
and often in the literature only piecewise constant elements are used.

<h3> Collocation boundary element method </h3>

By far, the most common approximation of boundary integral equations
is by use of the collocation based boundary element method.

This method requires the evaluation of the boundary integral equation
at a number of collocation points which is equal to the number of
unknowns of the system. The choice of these points is a delicate
matter, that requires a careful study. Assume that these points are
known for the moment, and call them $\mathbf x_i$ with $i=0...n\_dofs$.

The problem then becomes:
Given the datum $\mathbf{v}_\infty$, find a function $\phi_h$ in $V_h$
such that the following $n\_dofs$ equations are satisfied:

\f{align*}
    \alpha(\mathbf{x}_i) \phi_h(\mathbf{x}_i)
    - \int_{\Gamma_y} \frac{ \partial G(\mathbf{y}-\mathbf{x}_i)}{\partial\mathbf{n}_y }
    \phi_h(\mathbf{y}) \,ds_y =
    \int_{\Gamma_y} G(\mathbf{y}-\mathbf{x}_i) \,
    \mathbf{n}_y\cdot\mathbf{v_\infty} \,ds_y
    ,
\f}

where the quantity $\alpha(\mathbf{x}_i)$ is the fraction of (solid)
angle by which the point $\mathbf{x}_i$ sees the domain $\Omega$, as
explained above, and we set $\phi_\infty$ to be zero.  If the support
points $\mathbf{x}_i$ are chosen appropriately, then the problem can
be written as the following linear system:

\f[
\label{eq:linear-system}
(\mathbf{A}+\mathbf{N})\boldsymbol\phi = \mathbf{b},
\f]

where

\f[
\begin{aligned}
\mathbf{A}_{ij}&=
\alpha(\mathbf{x}_i) \psi_j(\mathbf{x}_i)
= \left (1+\int_\Gamma
\frac{\partial G(\mathbf{y}-\mathbf{x}_i)}{\partial \mathbf{n}_y}\,ds_y \right )
\psi_j(\mathbf{x}_i)
\\
\mathbf{N}_{ij}&= - \int_\Gamma
  \frac{\partial G(\mathbf{y}-\mathbf{x}_i)}{\partial \mathbf{n}_y}
  \psi_j(\mathbf{y}) \,ds_y
\\
\mathbf{b}_i&= \int_\Gamma
   G(\mathbf{y}-\mathbf{x}_i)  \, \mathbf{n}_y\cdot\mathbf{v_\infty}
   ds_y.
\end{aligned}
\f]

From a linear algebra point of view, the best possible choice of the
collocation points is the one that renders the matrix
$\mathbf{A}+\mathbf{N}$ the most diagonally dominant. A natural choice
is then to select the $\mathbf{x}_i$ collocation points to be the
support points of the nodal basis functions $\psi_i(\mathbf{x})$. In that
case, $\psi_j(\mathbf{x}_i)=\delta_{ij}$, and as a consequence the matrix
$\mathbf{A}$ is diagonal with entries
\f[
  \mathbf{A}_{ii}
  =
  1+\int_\Gamma
  \frac{\partial G(\mathbf{y}-\mathbf{x}_i)}{\partial \mathbf{n}_y}\,ds_y
  =
  1-\sum_j N_{ij},
\f]
where we have used that $\sum_j \psi_j(\mathbf{y})=1$ for the usual Lagrange
elements.
With this choice of collocation points, the computation of the entries
of the matrices $\mathbf{A}$, $\mathbf{N}$ and of the right hand side
$\mathbf{b}$ requires the evaluation of singular integrals on the
elements $K_i$ of the triangulation $\mathcal{T}_h$.
As usual in these cases, all integrations are performed on a reference
simple domain, i.e., we assume that each element $K_i$ of
$\mathcal{T}_h$ can be expressed as a linear (in two dimensions) or
bi-linear (in three dimensions) transformation of the reference
boundary element $\hat K \dealcoloneq [0,1]^{n-1}$, and we perform the integrations after a
change of variables from the real element $K_i$ to the reference
element $\hat K$.

<h3> Treating the singular integrals. </h3>

In two dimensions it is not necessary to compute the diagonal elements
$\mathbf{N}_{ii}$ of the system matrix, since, even if the denominator
goes to zero when $\mathbf{x}=\mathbf{y}$, the numerator is always
zero because $\mathbf{n}_y$ and $(\mathbf{y}-\mathbf{x})$ are
orthogonal (on our polygonal approximation of the boundary of $\Omega$), and
the only singular integral arises in the computation
of $\mathbf{b}_i$ on the i-th element of $\mathcal{T}_h$:
\f[
  \frac{1}{\pi}
  \int_{K_i}
  \ln|\mathbf{y}-\mathbf{x}_i| \, \mathbf{n}_y\cdot\mathbf{v_\infty} \,ds_y.
\f]

This can be easily treated by the QGaussLogR quadrature
formula.

Similarly, it is possible to use the QGaussOneOverR quadrature formula
to perform the singular integrations in three dimensions. The
interested reader will find detailed explanations on how these
quadrature rules work in their documentation.

The resulting matrix $\mathbf{A}+\mathbf{N}$ is full. Depending on its
size, it might be convenient to use a direct solver or an iterative
one. For the purpose of this example code, we chose to use only an
iterative solver, without providing any preconditioner.

If this were a production code rather than a demonstration of principles,
there are techniques that are available to not store full matrices but instead
store only those entries that are large and/or relevant. In the literature on
boundary element methods, a plethora of methods is available that allows to
determine which elements are important and which are not, leading to a
significantly sparser representation of these matrices that also facilitates
rapid evaluations of the scalar product between vectors and matrices. This not
being the goal of this program, we leave this for more sophisticated
implementations.


<h3>Implementation</h3>

The implementation is rather straight forward. The main point that hasn't been
used in any of the previous tutorial programs is that most classes in deal.II
are not only templated on the dimension, but in fact on the dimension of the
manifold on which we pose the differential equation as well as the dimension
of the space into which this manifold is embedded. By default, the second
template argument equals the first, meaning for example that we want to solve
on a two-dimensional region of two-dimensional space. The triangulation class
to use in this case would be <code>Triangulation@<2@></code>, which is an
equivalent way of writing <code>Triangulation@<2,2@></code>.

However, this doesn't have to be so: in the current example, we will for
example want to solve on the surface of a sphere, which is a two-dimensional
manifold embedded in a three-dimensional space. Consequently, the right class
will be <code>Triangulation@<2,3@></code>, and correspondingly we will use
<code>DoFHandler@<2,3@></code> as the DoF handler class and
<code>FE_Q@<2,3@></code> for finite elements.

Some further details on what one can do with things that live on
curved manifolds can be found in the report
<a target="_top"
href="http://www.dealii.org/reports/codimension-one/desimone-heltai-manigrasso.pdf"><i>Tools
for the Solution of PDEs Defined on Curved Manifolds with the deal.II
Library</i> by A. DeSimone, L. Heltai, C. Manigrasso</a>. In addition, the
step-38 tutorial program extends what we show here to cases where the equation
posed on the manifold is not an integral operator but in fact involves
derivatives.


<h3>Testcase</h3>

The testcase we will be solving is for a circular (in 2d) or spherical
(in 3d) obstacle. Meshes for these geometries will be read in from
files in the current directory and an object of type SphericalManifold
will then be attached to the triangulation to allow mesh refinement
that respects the continuous geometry behind the discrete initial
mesh.

For a sphere of radius $a$ translating at a velocity of $U$ in the $x$ direction, the potential reads
@f{align*}{
\phi = -\frac{1}{2}U \left(\frac{a}{r}\right)3 r \cos\theta
@f}
see, e.g. J. N. Newman, <i>Marine Hydrodynamics</i>, 1977,
pp. 127. For unit speed and radius, and restricting $(x,y,z)$ to lie
on the surface of the sphere,
$\phi = -x/2$. In the test problem,
the flow is $(1,1,1)$, so the appropriate exact solution on the
surface of the sphere is the superposition of the above solution with
the analogous solution along the $y$ and $z$ axes, or $\phi =
\frac{1}{2}(x + y + z)$.
