Fully implicit P0-based schemes for PDEs modeling phase change*

Abstract: We present our recent results on numerical schemes for evolution PDE models involving multivalued maximal monotone graphs. The unifying scheme of our work is fully implicit treatment of the nonlinearity and the use of lowest order spatial discretizations appropriate for the low regularity of the solutions and free boundaries, typical for the solutions of, e.g., parabolic variational inequalities or Stefan problem. The models we consider are typically part of larger coupled systems, and/or are subject to upscaling, thus stability, robustness, and conservation properties are key. We present some rigorous analyses and illustrate the use of the schemes for micro- and meso (Darcy) scale models arising in environmental studies linked to climate variability. These include phase change formation of biofilm and ice-water-grain phase transitions in permafrost, phase change materials, and formation/dissociation of hydrate crystals in subsea sediments.

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