Systematic construction of higher order bases for the finite element analysis of multiscale elliptic problems

• We propose a method for the construction of higher order MsFEM bases.
• The method is new, systematic, appropriate for parallel computations and numerically very efficient.
• We demonstrate the capabilities of the method through a set of examples.

We introduce a new approach to deriving higher order basis functions implemented in the Multiscale Finite Element Method (MsFEM) for elliptic problems. MsFEM relies on capturing small scale features of the system through bases utilized in the coarse scale solution. The proposed technique for the derivation of such bases is completely systematic and the increase in the associated computational cost is insignificant. We also show that the implementation of higher order bases in MsFEM leads to similar advantages as using higher order Lagrangian shape functions in the conventional finite element method. Various numerical examples for heat transfer problems with periodic or heterogeneous thermal properties are given to demonstrate the efficiency and improved characteristics of the proposed higher order bases.