Abstract
AbstractIn this paper, we present a two‐grid finite element method for the Allen‐Cahn equation with the logarithmic potential. This method consists of two steps. In the first step, based on a fully implicit finite element method, the Allen‐Cahn equation is solved on a coarse grid with mesh size H. In the second step, a linearized system whose nonlinear term is replaced by the value of the first step is solved on a fine grid with mesh size h. We give the energy stabilities of the traditional finite element method and the two‐grid finite element method. The optimal convergence order of the two‐grid finite element method in H1 norm is achieved when the mesh sizes satisfy h = O(H2). Numerical examples are given to demonstrate the validity of the proposed scheme. The results show that the two‐grid method can save the CPU time while keeping the same convergence rate.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callMore From: Numerical Methods for Partial Differential Equations
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
