# Fast Online Generalized Multiscale Finite Element Method using Constraint Energy Minimization

## TL;DR

This paper introduces an improved online multiscale finite element method that adaptively constructs basis functions in oversampled regions, achieving significant error reduction with minimal iterations and providing an adaptive algorithm with controllable convergence.

## Contribution

It presents a novel online basis construction approach using oversampling, enhancing error reduction and convergence in generalized multiscale finite element methods.

## Key findings

- Achieves three orders of magnitude error reduction.
- Error reduction is independent of physical parameters.
- Adaptive algorithm with user-defined convergence rate.

## Abstract

Local multiscale methods often construct multiscale basis functions in the offline stage without taking into account input parameters, such as source terms, boundary conditions, and so on. These basis functions are then used in the online stage with a specific input parameter to solve the global problem at a reduced computational cost. Recently, online approaches have been introduced, where multiscale basis functions are adaptively constructed in some regions to reduce the error significantly. In multiscale methods, it is desired to have only 1-2 iterations to reduce the error to a desired threshold. Using Generalized Multiscale Finite Element Framework, it was shown that by choosing sufficient number of offline basis functions, the error reduction can be made independent of physical parameters, such as scales and contrast. In this paper, our goal is to improve this. Using our recently proposed approach and special online basis construction in oversampled regions, we show that the error reduction can be made sufficiently large by appropriately selecting oversampling regions. Our numerical results show that one can achieve a three order of magnitude error reduction, which is better than our previous methods. We also develop an adaptive algorithm and enrich in selected regions with large residuals. In our adaptive method, we show that the convergence rate can be determined by a user-defined parameter and we confirm this by numerical simulations. The analysis of the method is presented.