}
+%-------------------------------------------------------------------------------
+% Step 34
+%-------------------------------------------------------------------------------
+
+@book{Newman1977,
+ author = {J. N. Newman},
+ title = {Marine Hydrodynamics},
+ journal = {},
+ year = 1977
+}
+
+
+@article{HeltaiBangerthKronbichler-2021,
+ author = {Luca Heltai and Wolfgang Bangerth and Martin Kronbichler and Andrea Mola},
+ journal = {Transactions on Mathematical Software},
+ month = {December},
+ number = {4},
+ pages = {1--30},
+ title = {Propagating geometry information to finite element computations},
+ volume = {47},
+ year = {2021}
+}
+
+
%-------------------------------------------------------------------------------
% Step 43
%-------------------------------------------------------------------------------
<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.
+Some further details on what one can do with things that live on curved
+manifolds can be found in the paper @cite HeltaiBangerthKronbichler-2021. 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>
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
+For a sphere of radius $a$ in dimension $d$, 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
+\phi = -\frac{1}{d-1} U \left(\frac{a^d}{r^{d-1}}\right) \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
+pp. 133 @cite Newman1977. For unit speed and radius, and restricting the point 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
+the flow is unitary in each coordinate direction, i.e., $(1,1)$ in dimension two and $(1,1,1)$ in dimension #3#, 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 $\phi =
-\frac{1}{2}(x + y + z)$.
+the analogous solutions along the other axes, or $\phi =
+(x + y)$ in dimension two and $\phi = \frac{1}{2}(x + y + z)$ in dimension three.
\ No newline at end of file