A moving mesh finite element method for Bernoulli free boundary problems

TL;DR

This work introduces a moving mesh finite element framework for Bernoulli free boundary problems by recasting the stationary problem as a MBP through pseudo-transient continuation. The MBP is solved in a split, time-marching fashion: update the free boundary with an Euler step, move interior nodes with the MMPDE moving mesh method, and solve the resulting IBVP on a linear FE mesh, using a quasi-Lagrangian formulation and a high-order Runge-Kutta time integrator. The approach yields a nonsingular, boundary-fitted mesh that remains valid for convex and concave domains and demonstrates second-order spatial accuracy with a quadratic gradient reconstruction, validated across exterior, interior, non-constant Bernoulli, and nonlinear FBPs. While robust and versatile, the method naturally converges slower than Newton-type methods, prompting discussions on speedups and future extensions to broader classes of free boundary problems.

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.

Figures (23)

• Figure 1: Illustration of the domain for exterior and interior Bernoulli FBPs.
• Figure 2: Illustration of boundary movement for MBP (\ref{['fbp-0']}).
• Figure 3: Example \ref{['fbp-ex1']}. The mesh of $N = 1998$ is plotted at $t=0$, 0.15, 0.3, and 0.456 for $\lambda = -2/\ln(0.6)$.
• Figure 4: Example \ref{['fbp-ex1']}. The maximum boundary velocity is plotted as a function of time for $\lambda = -2/\ln(0.6)$ and $N =1998$.
• Figure 5: Example \ref{['fbp-ex1']}. The error in the boundary location is plotted as a function of $N$ (the number of elements in the mesh) for two strategies (the quadratic least squares fitting and area-weighted averaging) for computing solution gradient used in boundary update.