Direct Problem for Gas Diffusion in Polar Firn with Variable Coefficients

TL;DR

This paper models gas trapping in polar firn using a degenerate parabolic PDE with variable coefficients, establishing existence, uniqueness, and numerical solutions through finite element methods.

Contribution

It extends previous models by allowing both diffusion and volume fraction coefficients to vary and degenerate, providing a rigorous mathematical and computational framework.

Findings

Proved existence and uniqueness of solutions for the degenerate PDE.

Developed a finite element Galerkin scheme for numerical approximation.

Established conditions for the well-posedness of the discrete system.

Abstract

We consider the mathematical model of gas trapping in deep polar ice (firns), which consists of a parabolic partial differential equation, that can degenerate at one boundary extreme. In [1], we considered all the coefficients to be constants, except the diffusion coefficient D(z) that is to be reconstructed. In this paper, we assume both the diffusion coefficient D(z) and the volume fraction f(z) are functions. The difficulty in this problem, both theoretically and computationally, arises from the fact that D(z) and f(z) may be zero at bottom of the firn. To handle such degeneracy, we defined appropriate weighted Sobolev spaces and used Lion's theorem to prove existence and uniqueness of the semi-variational formulation of the Firn PDE. A full discrete system is obtained through a P1 Finite element Galerkin procedure in space and an Euler-Implicit scheme in time. Sufficient conditions for the existence and uniqueness of the solution for the discrete system are obtained.