A Moving Mesh Finite Element Method for Bernoulli Free Boundary Problems (Communications in Computational Physics, 2024)
A moving mesh finite element method is studied for the numerical solutionof Bernoulli free boundary problems. The method is based on the pseudo-transientcontinuation 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 aremoved using a moving mesh method, and the corresponding initial-boundary valueproblem is solved using the linear finite element method. The method can take fulladvantages 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 domainfor a variety of geometries no matter if they are convex or concave. Moreover, it isconvergent towards steady state for a broad class of free boundary problems and initial guesses of the free boundary. Numerical examples for Bernoulli free boundaryproblems with constant and non-constant Bernoulli conditions and for nonlinear freeboundary problems are presented to demonstrate the accuracy and robustness of themethod and its ability to deal with various geometries and nonlinearities.
Keywords: Free boundary problem; moving boundary problem; moving mesh; finite element; pseudo-transient continuation.