Finite Element Method And Asymptotic Expansions For A Type Of Non-local Initial Boundary Value Problem

The paper mainly focuses on a type of non-local initial boundary valueproblem of the fnite element method and asymptotic expansions. Firstly,we introduces the current researches of non-local problems and give thefundamental theorem in this paper which will be used. Secondly, For thenon-local elliptic problems, we give the introduction of bilinear functionalA(·,·),space H1and fnite element subspace V h, and the operators P,Q. Inthe space H1, we prove that|P u|1is equivalent to∥u∥1. In the space V h, weprove that|Qu|1is equivalent to∥u∥1. we proposed numerical scheme whichis based on the fnite element method to solve the problem. We discussedthe posedness conditions for the numerical solution. We also obtain theasymptotic expansions for the numerical solution and super-convergenceresults of the derivative. In addition,some numerical examples are givento understand the correctness of the theory. Finally, we discussed nonlocalparabolic problems. We gives the Euler-Galerkin discretization and somenumerical examples to demonstrate the efectiveness of the method.