Adaptive Local Orthogonal Decomposition Method For Multiscale Elliptic Problems

In order to efficiently solve the problem of a class of second-order multi-scale ellip-tic,this paper developed an adaptive local orthogonal decomposition method.Local orthog-onal decomposition method is one of the effective method to solve problems of multi-scale numerical,its essence is to improve the traditional finite element method,and the specif-ic implementation is to modify the traditional finite element polynomial space.The main idea of this method is to put the multi-scale problems Weak solution space is decomposed into two orthogonal subspace.Them respectively called multi-scale space and kernel space interpolation operator,to the method used in numerical format structure of which is multi-scale subspace,it is a part of the micro information and low dimensional space.In order to construct the subspace,and often require in high dimensional kernel space to solve the equivalent of many of the original amount of calculation.In order to reduce the amount of computation,the subproblem needs to be localized and transformed into a saddle point problem.Finally,the subproblem is numerically solved under the sub-region of fine mesh subdivisionIn this paper,the specific implementation of the local orthogonal decomposition method is given,focus on the use of adaptive technology to improve the application efficiency of the method.Specific research work is to deduce the method based on H~1 norm a posteriori error estimation combined with the implementation process of the method and the posterior error results,three kinds of error estimators are defined,the algorithm flow of adaptive orthogonal decomposition method is designed,the local area size of numerical subproblem is automatically updated,the size of fine grid element for solving subproblem and the size of coarse grid element for solving original problem are automatically updated the numerical results show the effectiveness of the adaptive local orthogonal decomposition method.