The content describes a moving mesh finite element method for solving Bernoulli free boundary problems and other types of free boundary problems. The key highlights are:
The method is based on the pseudo-transient continuation approach, where a time-dependent moving boundary problem is constructed and solved until steady state is reached. The steady-state solution is then taken as the solution of the underlying free boundary problem.
The moving boundary problem is solved in a split manner at each time step: the moving boundary is updated using 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 moving mesh method, specifically the MMPDE (moving mesh PDE) method, is used to generate a new mesh for the updated domain at each time step. This method can handle domains of various geometries, including convex and concave shapes, and guarantees a non-singular mesh.
Two approaches, area-averaging and quadratic least squares fitting, are discussed for reconstructing the gradient of the finite element solution and the normal to the boundary at the boundary vertices, which are needed for the Bernoulli condition but not directly available in the standard finite element approximation.
Numerical examples are presented for Bernoulli free boundary problems with constant and non-constant Bernoulli conditions, as well as for nonlinear free boundary problems involving the p-Laplacian and an obstacle problem. The results demonstrate the accuracy, robustness, and flexibility of the proposed moving mesh finite element method.
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by Jinye Shen,H... at arxiv.org 04-09-2024
https://arxiv.org/pdf/2404.04418.pdfDeeper Inquiries