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Error estimates for a finite element discretization of the Cahn-Hilliard-Gurtin equations. (English) Zbl 1227.65080
The authors consider the Cahn-Hilliard-Gurtin equations
\begin{aligned} \partial_t u - a\cdot \nabla \partial_t u &= \text{div}(B \nabla w) + m \quad \text{in } \Omega \times (0,\infty),\\ w-b \cdot \nabla w &= \beta \partial_t u - \alpha \Delta u +f'(u)-\gamma \quad \text{in } \Omega \times (0,\infty),\\ u(0) &= u_0\end{aligned}
in a parallelepiped $$\Omega \subset \mathbb{R}^d$$ subject to periodic boundary conditions. It is assumed that the coercivity condition
$\beta x^2+y^tBy+y^t(a+b)x \geq c_0(x^2+\| y \|^2)$
holds, where $$c_0>0$$ and $$\| y \|$$ denotes the Euclidean norm in $$\mathbb{R}^d$$. The source $$m=m(t)$$ is assumed to have vanishing mean value for all $$t \in [0,T]$$. The equations model features of two-phase systems, where $$u$$ is the order parameter and $$w$$ is the chemical potential.
The authors prove an energy a priori estimate and, consequently, existence and uniqueness of a weak solution of the given problem in suitable Sobolev spaces. Next, the problem is semi-discretized using linear finite elements and optimal error estimates are proved. Also the case of reduced regularity assumptions for $$u$$ and $$w$$ is considered. A corresponding error analysis is given for the fully discrete problem, which is obtained by discretizing the time derivative using the first order backward differentiation formula.
For the fully discrete problem without source term it is shown that the discrete solution converges to equilibrium as time goes to infinity. Numerical examples with $$d=1$$ and $$d=2$$ are provided.

##### MSC:
 65M15 Error bounds for initial value and initial-boundary value problems involving PDEs 65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs 65M20 Method of lines for initial value and initial-boundary value problems involving PDEs 35K55 Nonlinear parabolic equations 35Q35 PDEs in connection with fluid mechanics