TY - JOUR
T1 - A New Two-Grid Method for Expanded Mixed Finite Element Solution of Nonlinear Reaction Diffusion Equations
AU - Liu , Shang
AU - Chen , Yanping
JO - Advances in Applied Mathematics and Mechanics
VL - 3
SP - 757
EP - 774
PY - 2018
DA - 2018/05
SN - 9
DO - http://doi.org/10.4208/aamm.2015.m1370
UR - https://global-sci.org/intro/article_detail/aamm/12174.html
KW - Error estimation, mixed finite elements, reaction-diffusion equations, two-grid methods.
AB - <p style="text-align: justify;">In the paper, we present an efficient two-grid method for the approximation
of two-dimensional nonlinear reaction-diffusion equations using an expanded mixed
finite-element method. We transfer the nonlinear reaction diffusion equation into first
order nonlinear equations. The solution of the nonlinear system on the fine space is
reduced to the solutions of two small (one linear and one non-linear) systems on the
coarse space and a linear system on the fine space. Moreover, we obtain the error estimation for the two-grid algorithm. It is showed that coarse space can be extremely
coarse and achieve asymptotically optimal approximation as long as the mesh sizes
satisfy $h^{k+1}=\mathcal{O}(H^{3k+1})$. A numerical example is also given to illustrate the effectiveness of the algorithm.</p>