Numerical study of the porous medium equation with absorption, variable exponents of nonlinearity and free boundary

摘要

In this paper, we study an application of the moving mesh method to the porous medium equation with absorption and variable exponents of nonlinearity in 2D domains with moving boundaries. The boundary's movement is governed by an equation prompted by the Darcy law and the spatial discretization is defined by a triangulation of the domain. At each finite element, the solution is approximated by piecewise polynomial functions of degree r ≥ 1 using Lagrange interpolating polynomials in area coordinates. The vertices of the triangles move according to a system of differential equations which is added to the equations of the problem. The resulting system is converted into a system of ordinary differential equations in time variable, which is solved using a suitable integrator. The integrals that arise in the system of ordinary differential equations are calculated using the Gaussian quadrature. Finally, we present some numerical results of application of this technique.