A moving mesh finite element method for Bernoulli free boundary problems (2404.04418v1)
Abstract: 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.
- H. W. Alt and L. A. Cafarelli. Existence and regularity for a minimum problem with free boundary. J. Reine Angew. Math. 325 (1981), 105-144.
- P. Cardaliaguet and R. Tahraoui. Some uniqueness results for Bernoulli interior free-boundary problems in convex domains. Elec. J. Diff. Eq. 2002 (2002), 1-16.
- Y. Du and Z. G. Lin. Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary. SIAM J. Math. Anal. 42 (2010), 377-405.
- K. Eppler and H. Harbrecht. Efficient treatment of stationary free boundary problems. Appl. Numer. Math. 56 (2006), 1326-1339.
- K. Eppler and H. Harbrecht. Shape optimization for free boundary problems - analysis and numerics. Int. Ser. Numer. Math. 160 (2012), 277-288.
- A. Friedman and B. Hu. Asymptotic stability for a free boundary problem arising in a tumor model. J. Diff. Eq. 227 (2006), 598-639.
- A. Henrot and M. Onodera. Hyperbolic solutions to Bernoulli’s free boundary problem. Arch. Rational Mech. Anal. 240 (2021), 761-784.
- A. Henrot and H. Shahgholian. The one phase free boundary problem for the p𝑝pitalic_p-Laplacian with non-constant Bernoulli boundary condition. Trans. Amer. Math. Soc. 354 (2002), 2399-2416.
- W. Huang. An introduction to MMPDElab. arXiv:1904.05535.
- W. Huang and L. Kamenski. A geometric discretization and a simple implementation for variational mesh generation and adaptation. J. Comput. Phys. 301 (2015), 322-337.
- W. Huang and L. Kamenski. On the mesh nonsingularity of the moving mesh PDE method. Math. Comp. 87 (2018), 1887-1911.
- C. M. Murea and G. Hentschel. Finite element methods for investigating the moving boundary problem in biological development. Prog. Nonlin. Diff. Eq. Appl. 64 (2005), 357-371.
- C. Ngo and W. Huang. Adaptive finite element solution of the porous medium equation in pressure formulation. Numer. Meth. Part. Diff. Eq. 35 (2019), 1224-1242.
- R. Rangarajan and A. J. Lew. Analysis of a method to parameterize planar curves immersed in triangulations. SIAM J. Numer. Anal. 51 (2013), 1392-1420.
- R. Rangarajan and A. J. Lew. Universal meshes: A method for triangulating planar curved domains immersed in nonconforming triangulations. Int. J. Numer. Meth. Eng. 98 (2014), 236-264.
- J. F. T. Rabago. Analysis and numerics of novel shape optimization methods for the Bernoulli problem. Nagoya University, 2020. PhD Thesis.
- Y. Yamada. Asymptotic properties of a free boundary problem for a reaction-diffusion equation with multi-stable nonlinearity. Rend. Istit. Mat. Univ. Trieste 52 (2020), 65-89.
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