Nonnegativity preserving convergent schemes for stochastic porous-medium equations

Grillmeier, Hubertus; Grün, Günther

doi:10.1090/mcom/3372

Nonnegativity preserving convergent schemes for stochastic porous-medium equations
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Math. Comp. 88 (2019), 1021-1059 Request permission

Abstract:

We propose a fully discrete finite-element scheme for stochastic porous-medium equations with linear, multiplicative noise given by a source term. A subtle discretization of the degenerate diffusion coefficient combined with a noise approximation by bounded stochastic increments permits us to prove $H^1$-regularity and nonnegativity of discrete solutions. By Nikol′skiĭ estimates in time, Skorokhod-type arguments and the martingale representation theorem, convergence of appropriate subsequences towards a weak solution is established. Finally, some preliminary numerical results are presented which indicate that linear, multiplicative noise in the sense of Ito, which enters the equation as a source-term, has a decelerating effect on the average propagation speed of the boundary of the support of solutions.

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