A Study on Moving Mesh Finite Element Solution of the Porous Medium Equation
For researchers in numerical PDEs, this is an incremental improvement applying a known method (moving mesh FEM) to a specific equation class.
The paper studies an adaptive moving mesh finite element method for the porous medium equation, achieving first-order convergence with uniform and arclength-based meshes and second-order convergence with Hessian-based adaptive meshes, and demonstrates its applicability to complex free boundaries and variable exponents.
An adaptive moving mesh finite element method is studied for the numerical solution of the porous medium equation with and without variable exponents and absorption. The method is based on the so-called moving mesh partial differential equation approach and employs its newly developed implementation. Three types of mesh are considered, uniform and arclength-based and Hessian-based adaptive meshes. The method shows a first order convergence for uniform and arclength-based adaptive meshes and a second-order convergence for Hessian-based adaptive meshes. It is also shown that the method can be used for situations with complex free boundaries, emerging and splitting of free boundaries, and the porous medium equation with variable exponents and absorption. Two dimensional numerical results are presented.