TY - JOUR
T1 - Numerical Continuation and Bifurcation for Differential Geometric PDEs
AU - Meiners , Alexander
AU - Uecker , Hannes
JO - Numerical Mathematics: Theory, Methods and Applications
VL - 3
SP - 555
EP - 606
PY - 2024
DA - 2024/08
SN - 17
DO - http://doi.org/10.4208/nmtma.OA-2024-0005
UR - https://global-sci.org/intro/article_detail/nmtma/23367.html
KW - Numerical bifurcation, constant mean curvature, Helfrich functional, discrete differential geometry.
AB - <p style="text-align: justify;">We describe some differential geometric bifurcation problems and their
treatment in the Matlab continuation and bifurcation toolbox pde2path. The continuation steps consist in solving the PDEs for the normal displacement of an immersed surface $X ⊂\mathbb{R}^3,$ with bifurcation detection and possible subsequent branch
switching. The examples include minimal surfaces such as Enneper’s surface and
a Schwarz-P-family, some non-zero constant mean curvature surfaces such as liquid
bridges, and some 4th order biomembrane models. In all of these we find interesting
symmetry-breaking bifurcations. A few of these are (semi)analytically known and
hence used as benchmarks.</p>