DCDS-B
Two novel decoupling algorithms for the steady Stokes-Darcy model based on two-grid discretizations
Tong Zhang Jinyun Yuan
Discrete & Continuous Dynamical Systems - B 2014, 19(3): 849-865 doi: 10.3934/dcdsb.2014.19.849
In this work, two novel decoupling algorithms for the steady Stokes-Darcy model based on two-grid discretizations are proposed and analyzed. Optimal error estimates for these variables are presented. Two grid decoupled scheme proposed by Mu and Xu (2007) is used to develop the two novel decoupling algorithms. For Algorithm 3.2, the optimal error estimates are obtained for both ${\bf{u}}_f,\ p_f$ and $\phi$ with mesh sizes satisfying $H=\sqrt{h}$. For Algorithm 3.3, the convergence of $\phi$ in $H^1$-norm is improved form $H^2$ to $H^\frac{5}{2}$. Furthermore, the existing results in [17] are improved and supplemented. Finally, some numerical experiments are provided to show the efficiency and effectiveness of the developed algorithms.
keywords: Stokes equations Darcy's law multimodeling problems two grid method.
DCDS-B
The stabilized semi-implicit finite element method for the surface Allen-Cahn equation
Xufeng Xiao Xinlong Feng Jinyun Yuan
Discrete & Continuous Dynamical Systems - B 2017, 22(7): 2857-2877 doi: 10.3934/dcdsb.2017154

Two semi-implicit numerical methods are proposed for solving the surface Allen-Cahn equation which is a general mathematical model to describe phase separation on general surfaces. The spatial discretization is based on surface finite element method while the temporal discretization methods are first-and second-order stabilized semi-implicit schemes to guarantee the energy decay. The stability analysis and error estimate are provided for the stabilized semi-implicit schemes. Furthermore, the first-and second-order operator splitting methods are presented to compare with stabilized semi-implicit schemes. Some numerical experiments including phase separation and mean curvature flow on surfaces are performed to illustrate stability and accuracy of these methods.

keywords: Surface Allen-Cahn equation surface finite element method stabilized semi-implicit scheme operator splitting method error estimate

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