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This paper studies finite element approximations of the stochastic Allen--Cahn equation with gradient-type multiplicative noise that is white in time and correlated in space. The sharp interface limit--as the diffuse interface thickness vanishes--of the stochastic Allen--Cahn equation is formally a stochastic mean curvature flow which is described by a stochastically perturbed geometric law of the deterministic mean curvature flow. Two fully discrete finite element methods which are based on different time-stepping strategies for the nonlinear term are proposed. Strong convergence with sharp rates for both fully discrete finite element methods is proved. This is done with the crucial help of the Holder continuity in time with respect to the spatial L2-norm and H¹-seminorm for the strong solution of the stochastic Allen--Cahn equation. It also relies on the fact that high moments of the strong solution are bounded in various spatial and temp oral norms. Numerical experiments are provided to gauge the performance of the proposed fully discrete finite element methods and to study the interplay of the geometric evolution and gradient-type noise. [ABSTRACT FROM AUTHOR]

This paper proposes and analyzes two fully discrete mixed interior penalty discontinuous Galerkin (DG) methods for the fourth order nonlinear Cahn-Hilliard equation. Both methods use the backward Euler method for time discretization and interior penalty DG methods for spatial discretization. They differ from each other on how the nonlinear term is treated; one of them is based on fully implicit time-stepping and the other uses energy-splitting time-stepping. The primary goal of the paper is to prove the convergence of the numerical interfaces of the DG methods to the interface of the Hele-Shaw flow. This is achieved by establishing error estimates that depend on ϵ-1 only in some low polynomial orders, instead of exponential orders. Similar to [X. Feng and A. Prohl, Numer. Math., 74 (2004), pp. 47-84], the crux is to prove a discrete spectrum estimate in the discontinuous Galerkin finite element space. However, the validity of such a result is not obvious because the DG space is not a subspace of the (energy) space HM1(Ω) and it is larger than the finite element space. This difficulty is overcome by a delicate perturbation argument which relies on the discrete spectrum estimate in the finite element space proved by Feng and Prohl. Numerical experiment results are also presented to gauge the theoretical results and the performance of the proposed fully discrete mixed DG methods. [ABSTRACT FROM AUTHOR]

This paper develops and analyses two fully discrete interior penalty discontinuous Galerkin (IP-DG) methods for the Allen–Cahn equation, which is a nonlinear singular perturbation of the heat equation and originally arises from phase transition of binary alloys in materials science, and its sharp interface limit (the mean curvature flow) as the perturbation parameter tends to zero. Both fully implicit and energy-splitting time-stepping schemes are proposed. The primary goal of the paper is to derive sharp error bounds which depend on the reciprocal of the perturbation parameter ∈ (also called 'interaction length') only in some lower polynomial order, instead of exponential order, for the proposed IP-DG methods. The derivation is based on a refinement of the nonstandard error analysis technique first introduced in Feng & Prohl (2003, Numerical analysis of the Allen–Cahn equation and approximation for mean curvature flows. Numer. Math., 2003, 94, 33–65). The centrepiece of this new technique is to establish a spectrum estimate result in totally discontinuous DG finite element spaces with the help of a similar spectrum estimate result in the conforming finite element spaces which was established in Feng & Prohl. As a nontrivial application of the sharp error estimates, they are used to establish convergence, and the rates of convergence of the zero-level sets of the fully discrete IP-DG solutions to the classical and generalized mean curvature flow. Numerical experiment results are also presented to gauge the theoretical results and the performance of the proposed fully discrete IP-DG methods. [ABSTRACT FROM AUTHOR]

This paper develops a posteriori error estimates of residual type for conforming and mixed finite element approximations of the fourth order Cahn-Hilliard equation ut + Δ(εΔu - ε-1 if(u)) = 0. It is shown that the a posteriori error bounds depends on ε-1 only in some low polynomial order, instead of exponential order. Using these a posteriori error estimates, we construct an adaptive algorithm for computing the solution of the Cahn-Hilliard equation and its sharp interface limit, the Hele-Shaw flow. Numerical experiments are presented to show the robustness and effectiveness of the new error estimators and the proposed adaptive algorithm. [ABSTRACT FROM AUTHOR]