Deffiecient Computation Of Multiscale Finite Element Basis Space For Elliptic Equations With Perturbed Coefficients Using Its Sparsity

This article aims to explore a new numerical method for solving configuration which is common as a class of elliptic partial differential equations multiscale problems.Traditional means to solve such problems is to use the classical multiscale finite element method and finite element method.And for such specific equation there spawned another two construction methods based on Rough Polyharmonic Splines(RPS): One is the Global Rough Polyharmonic Splines(G-RPS),another is the Local Rough Polyharmonic Splines(L-RPS).These methods can work out the numerical solution well under the condition of saving computing resources and ensuring accuracy modest circumstances.But if we make some small local disturbance factors for such multi-scale problems,we hope we can be able to quickly direct the original problems and the impact of these disturbances on the load to construct a new numerical solution of multiscale problems,rather than recalculation again.This requires taking into account large-scale sparse nature of the resulting disturbances for such factors.In this paper,I have made adequate preparation of exploratory work for such ideas to construct a new numerical method.And eventually find that when the original multi-scale coefficient perturbations occur,there actually exists a low rank between the original problem and the numerical solution of new problems;furthermore,factors for this low rank is clear.Therefore,this idea from this article has showed its effectiveness and efficiency.On the other hand,a lot of work is needed for further exploration.