Direct Problem for Gas Diffusion in Polar Firn with Variable Coefficients
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| Format: | Preprint |
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2026
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| _version_ | 1866908989778821120 |
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| author | Moufawad, Sophie Nassif, Nabil Triki, Faouzi |
| author_facet | Moufawad, Sophie Nassif, Nabil Triki, Faouzi |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_21821 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Direct Problem for Gas Diffusion in Polar Firn with Variable Coefficients Moufawad, Sophie Nassif, Nabil Triki, Faouzi Numerical Analysis Analysis of PDEs 35A01, 35A02, 35A15, 35G16, 35M13, 65M22, 65M60 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. |
| title | Direct Problem for Gas Diffusion in Polar Firn with Variable Coefficients |
| topic | Numerical Analysis Analysis of PDEs 35A01, 35A02, 35A15, 35G16, 35M13, 65M22, 65M60 |
| url | https://arxiv.org/abs/2604.21821 |