Application of the moving mesh method to the porous medium equation with variable exponent

In this paper, we investigate the applicability of a moving mesh method when applied to the porous medium equation with variable exponent in 2D with free boundaries.The movement of the boundary is derived by a discretization of the Darcy law. The spatial discretization is defined by a partition in triangles of the domain, and the solution is approximated by piecewise polynomial functions of degree r≥1r≥1 using Lagrange interpolating polynomials in area coordinates. A parabolic system of differential equations to control the movement of the interior vertices of the triangles is added to the equations of the problem. The resulting ordinary differential system of equations in time variable is solved using a suitable integrator. The integrals that arise in the system of ordinary differential equations are calculated using the Gaussian quadrature of appropriate order. Finally, we present a numerical application of this technique and compare the results with a fixed mesh method.