From af4207c1ae63365011264c06e97d92552e4dfb32 Mon Sep 17 00:00:00 2001 From: Luca Heltai Date: Thu, 20 Feb 2025 16:10:30 +0100 Subject: [PATCH] Fix exact solution step-34. --- doc/doxygen/references.bib | 24 ++++++++++++++++++++++++ examples/step-34/doc/intro.dox | 26 +++++++++++--------------- 2 files changed, 35 insertions(+), 15 deletions(-) diff --git a/doc/doxygen/references.bib b/doc/doxygen/references.bib index 09314c7573..02a9be6983 100644 --- a/doc/doxygen/references.bib +++ b/doc/doxygen/references.bib @@ -370,6 +370,30 @@ } +%------------------------------------------------------------------------------- +% 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 %------------------------------------------------------------------------------- diff --git a/examples/step-34/doc/intro.dox b/examples/step-34/doc/intro.dox index 0c18932075..6e46d1528d 100644 --- a/examples/step-34/doc/intro.dox +++ b/examples/step-34/doc/intro.dox @@ -653,15 +653,11 @@ will be Triangulation@<2,3@>, and correspondingly we will use DoFHandler@<2,3@> as the DoF handler class and FE_Q@<2,3@> for finite elements. -Some further details on what one can do with things that live on -curved manifolds can be found in the report -Tools -for the Solution of PDEs Defined on Curved Manifolds with the deal.II -Library by A. DeSimone, L. Heltai, C. Manigrasso. 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.

Testcase

@@ -673,15 +669,15 @@ 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 +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, Marine Hydrodynamics, 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 -- 2.39.5