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.

翻译：研究了一种求解伯努利自由边界问题数值解的动网格有限元方法。该方法基于拟瞬态延拓技术，通过构造移动边界问题，并将其稳态解作为原始伯努利自由边界问题的解。在每个时间步中，移动边界问题以分裂方式求解：采用欧拉格式更新移动边界，利用动网格方法移动内部网格节点，并应用线性有限元法求解相应的初边值问题。该方法能够充分利用拟瞬态延拓与动网格技术的双重优势，特别是可避免网格缠绕，适用于凸、凹等各类几何形状的变域适配。此外，该方法对广泛的自由边界问题及自由边界初始猜测均具有向稳态收敛的特性。通过常/非常数伯努利条件及非线性自由边界问题的数值算例，验证了该方法在精度、鲁棒性及处理多类几何构型与非线性特征方面的能力。