A C~0-Weak Galerkin Method For Stochastic Cahn-Hilliard-Cook Equation

In this paper,we study a numerical computational method for approximating a class of stochastic Cahn-Hilliard-Cook equations using a C~0weak finite element method.How-ever,due to the low regularity of the solution process of stochastic differential equations,it is hard to derive the convergence property of the numerical solution.The stochastic Cahn-Hilliard equation was studied as early as 1990s,Da Prato et al.studied the ex-istence and uniqueness of solutions for the stochastic Cahn-Hilliard equation driven by additive white noise.Later,Cardon extended this result to the Cahn-Hilliard equation driven by nonlinear Gaussian noise.Stig Larsson et al.studied the numerical approxima-tion scheme of the linear stochastic Cahn-Hilliard-Cook equation via the classical finite element method and derived the convergence property.As we know,for the equation whose spatial variable is fourth-order,its conforming finite element space should be combined with C~1-smoothness elements.For example,the degree of a polynomial in the Argris element is at least 5,and is 21 in the two-dimensional case.Compared with the existing results,the advantage of the continuous weak finite element method is that it avoids the construction of a complex C~1element finite element space and thus greatly reduces the number of degrees of freedom of the space base in the calculation.In this thesis,we approximate the classical Laplacian operator by the weak Laplacian operator,discretize the random Cahn-Hilliard-Cook spatial variables using the C~0weak Galerkin finite element method,and provide the optimal error order estimates.At the same time,the classical backward Euler scheme is used to discretize the time variables.At the end of this paper,the numerical examples are given to verify the validity and consistency of the C~0weak finite element method for solving the linear stochastic Cahn-Hilliard-Cook equation.