Adaptive Finite Element Algorithm Based On A Posterior Error Estimation

In engineering and scientific computing,partial differential equations can be used to describe many practical problems.However,most partial differential equations have no or difficult to obtain accurate solutions,and can only be approximated by appropriate numerical methods.Unfortunately,umerical calculations generally produce errors.Therefore,not only the approximate solution is important,but the error analysis is also very important,that is,the norm of the difference between the exact solution and the approximate solution.The exact error is often unknown,which means that we can only obtain some estimates of the error.In many cases,to achieve efficiency and computable a posterior error estimates is very difficult.While the adaptive algorithm has a strong dependence on the a posterior error estimation.This type of algorithm is based on the calculation result of the a posterior error estimation,selecting the best discrete method,and using the least amount of calculation to obtain a higher accuracy solution.Adaptive finite element algorithm has been developed rapidly in recent ten years.Because it can automatically find the largest error structure and process it with mesh refinement technology according to the a posterior error estimation.It also effectively improve the calculation efficiency and the accuracy of the solution is improved by the obtained error.Therefore,it has been highly praised by engineers.We take the representative elliptic equation as the research object.Firstly,two a posterior error estimates based on the hierarchical base type and the recovery type are given separately.Through numerical experiments,we found that the result of any kind a posterior error estimate may be biased and affect the conforming of the mesh.Therefore,we use the weighted average of the two a posterior error estimates obtained above as the new a posterior error estimator,and appropriately increase the error threshold of the mesh node that needs to be refined.And the relevant adaptive finite element algorithm is given.In the process of adaptation,we use Dorfler appropriately increase the error threshold of the mesh node that needs to be refined.And the relevant adaptive finite element algorithm is given.In the process of adaptation,we use Dorfler criterion and the latest vertex bisection mesh refinement technology to adjust the mesh,and use examples with Dirichlet boundary conditions to verify the proposed algorithm.The results of the numerical examples show that,compared with the uniform mesh,the adaptive algorithm automatically refines the mesh where the solution change is relatively large,and automatically coarses the mesh where the solution change is relatively gentle.The conforming of the mesh has been improved at the same time.We could obtain higher-precision calculation results while the calculation efficiency is high.Which illustrates the effectiveness and reliability of this algorithm for solving elliptic equations.