Finite elements for Wasserstein $W_p$ gradient flows
Résumé
Convergence of a finite element discretization of a degenerate parabolic equation
of $q$-Laplace type with an additional external potential is considered.
The main novelty of our approach is that we use the underlying gradient flow structure
in the $L^p$-Wasserstein metric:
from the abstract machinery of metric gradient flows,
convergence of scheme is obtained solely on the basis of estimates
that result naturally from the equation's variational structure.
In particular, the limit is identified as the unique gradient flow solution
without reference to monotonicity methods.
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