Research On Adaptive Grid Algorithm For Reaction-Diffusion Equations

In scientific and engineering calculation,there are a lot of practical problems which can be expressed by partial differential equations.But for the numerical solution of practical problems of physics or engineering,the numerical approximation solutions usually have very large errors due to the singularities of the local region,such as interior or boundary layers,or sharp shock-like fronts.For the calculation of this kind of problem,.if adopt uniform subdivision,it is necessary to divide the meshes very densely and the cost of the computation may increase substantially.In order to improve the precision of the solution without increasing the computation,the adaptive mesh method is a pretty natural choice.When solving model problem by the adaptive mesh method,the meshes automatically refine the grids in the region where the solution changes dramatically,while relatively sparse in the region where the solution changes smoothly.It can obtain high-precision solutions while maintaining high computational efficiency.Taking a Poisson equation as an example firstly,we provide the adaptive mesh algorithm based on the residual and recovery a posterior error estimates,and obtain the corresponding adaptive algorithm.Two examples with Dirichlet boundary conditions are provided in order to illustrate the algorithm.The results show that the grid is automatically refined at the place where the gradient of the solution is large,and the effect of the result based on recovery-type error indicator is better.Then,the adaptive mesh method is applied to solve the reaction-diffusion problem.Firstly,theoretical analysis results were given Then,based on the recovery-type posterior error estimation indicator,Dorfler criterion and red-green refining strategy are used,we propose the corresponding adaptive time-space finite element algorithm.Finally,based on the above adaptive methods,a class of reaction-diffusion problems with initial and boundary conditions are given to verify the results.The results on the coarse mesh and the uniformly refined mesh are also provided for comparison.The numerical results show that,compared with the uniform refined grids,the adaptive grids can automatically refine at the place where the solution the gradient is larger.With less computation,more accurate results are achieved,which illustrate the effectiveness of the proposed algorithm for reaction-diffusion problems.