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Computational analysis demonstrates a stable finite element scheme for advection-diffusion on moving curves under stochastic noise, indicating robust numerical convergence for rough forcing.
In this paper we consider an evolving surface finite element method method for the advection and diffusion of a scalar quantity on a moving closed curve. The diffusion process is controlled by a forcing term that may include a rough term (specifically a stochastic noise) which in particular destroys the classical time differentiability properties of the solution. We provide a suitable variational solution concept and a fully discrete finite element method discretization. Our error analysis appropriately generalizes classical estimates to this weaker setting. We present some numerical simulations that confirm our theoretical findings.
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Pozzi et al. (2026) studied this question.
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