TY - JOUR
T1 - A Moving Mesh Finite Element Method for Bernoulli Free Boundary Problems
AU - Shen , Jinye
AU - Dai , Heng
AU - Huang , Weizhang
JO - Communications in Computational Physics
VL - 1
SP - 248
EP - 273
PY - 2024
DA - 2024/07
SN - 36
DO - http://doi.org/10.4208/cicp.OA-2023-0214
UR - https://global-sci.org/intro/article_detail/cicp/23303.html
KW - Free boundary problem, moving boundary problem, moving mesh, finite element, pseudo-transient continuation.
AB - <p style="text-align: justify;">A moving mesh finite element method is studied for the numerical solution
of Bernoulli free boundary problems. The method is based on the pseudo-transient
continuation with which a moving boundary problem is constructed and its steady-state solution is taken as the solution of the underlying Bernoulli free boundary problem. The moving boundary problem is solved in a split manner at each time step:
the moving boundary is updated with the Euler scheme, the interior mesh points are
moved using a moving mesh method, and the corresponding initial-boundary value
problem is solved using the linear finite element method. The method can take full
advantages of both the pseudo-transient continuation and the moving mesh method.
Particularly, it is able to move the mesh, free of tangling, to fit the varying domain
for a variety of geometries no matter if they are convex or concave. Moreover, it is
convergent towards steady state for a broad class of free boundary problems and initial guesses of the free boundary. Numerical examples for Bernoulli free boundary
problems with constant and non-constant Bernoulli conditions and for nonlinear free
boundary problems are presented to demonstrate the accuracy and robustness of the
method and its ability to deal with various geometries and nonlinearities.</p>