Finite element approximation of an anisotropic porous medium equation with fractional pressure

TL;DR

This paper develops and analyzes a finite element method for a nonlocal, anisotropic porous medium equation involving fractional elliptic operators, proving convergence and demonstrating numerical behavior.

Contribution

The paper introduces a convergent finite element scheme for a fractional anisotropic porous medium equation with spectral fractional pressure, including numerical experiments.

Findings

Convergence of the finite element scheme to a weak solution.

Numerical experiments illustrate nonlocal effects and anisotropy.

Failure of the comparison principle observed numerically.

Abstract

We study a nonlocal diffusion equation of porous medium type featuring a generalised fractional pressure with spatial anisotropy. We construct a finite element method for the numerical solution of the equation on a bounded open Lipschitz polytopal domain $\Omega \subset \mathbb{R}^{d}$, where $d = 2$ or $3$. The pressure in the model is defined as the solution of fractional elliptic problem involving the fractional power of a second order differential operator, in terms of its spectral definition. Under suitable assumptions on the fractional order and the coefficients of the operator, we rigorously prove convergence of the numerical scheme. The analysis is carried out in two stages: first passing to the limit in the spatial discretization, and then in the time step, ultimately showing that a subsequence of the sequence of finite element approximations defined by the proposed numerical method converges to a bounded and nonnegative weak solution of the initial-boundary-value problem under consideration. Finally, we present numerical experiments in two dimensions illustrating the computational aspects of the method and highlighting the interplay between nonlocal effects and spatial anisotropy under different configurations. We also show numerically the failure of the comparison principle and exponential decay of the numerical solution to a steady state.