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• W2945575610 abstract "Abstract We consider the stochastic Cahn–Hilliard equation with additive noise term $$varepsilon ^gamma g, {dot{W}}$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:msup> <mml:mi>ε</mml:mi> <mml:mi>γ</mml:mi> </mml:msup> <mml:mi>g</mml:mi> <mml:mspace /> <mml:mover> <mml:mi>W</mml:mi> <mml:mo>˙</mml:mo> </mml:mover> </mml:mrow> </mml:math> ( $$gamma >0$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>γ</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> ) that scales with the interfacial width parameter $$varepsilon $$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mi>ε</mml:mi> </mml:math> . We verify strong error estimates for a gradient flow structure-inheriting time-implicit discretization, where $$varepsilon ^{-1}$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:msup> <mml:mi>ε</mml:mi> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:math> only enters polynomially ; the proof is based on higher-moment estimates for iterates, and a (discrete) spectral estimate for its deterministic counterpart. For $$gamma $$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mi>γ</mml:mi> </mml:math> sufficiently large, convergence in probability of iterates towards the deterministic Hele–Shaw/Mullins–Sekerka problem in the sharp-interface limit $$varepsilon rightarrow 0$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>ε</mml:mi> <mml:mo>→</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> is shown. These convergence results are partly generalized to a fully discrete finite element based discretization. We complement the theoretical results by computational studies to provide practical evidence concerning the effect of noise (depending on its ’strength’ $$gamma $$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mi>γ</mml:mi> </mml:math> ) on the geometric evolution in the sharp-interface limit. For this purpose we compare the simulations with those from a fully discrete finite element numerical scheme for the (stochastic) Mullins–Sekerka problem. The computational results indicate that the limit for $$gamma ge 1$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>γ</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> is the deterministic problem, and for $$gamma =0$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>γ</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> we obtain agreement with a (new) stochastic version of the Mullins–Sekerka problem." @default.
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• W2945575610 date "2021-02-17" @default.
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• W2945575610 title "Numerical approximation of the stochastic Cahn–Hilliard equation near the sharp interface limit" @default.
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