Finite element solution of multi-scale transport problems using the least squares based bubble function enrichment

TL;DR

This paper introduces a least squares based method to derive bubble functions for enriching finite elements, improving accuracy in multi-scale transport problems without increasing computational cost.

Contribution

It develops an optimal least squares approach for bubble function derivation that enhances finite element solutions without additional degrees of freedom.

Findings

Enriched elements outperform standard linear elements in accuracy.

The method achieves this without extra computational cost.

Higher order enrichments are needed for strongly convection-dominated problems.

Abstract

This paper presents an optimum technique based on the least squares method for the derivation of the bubble functions to enrich the standard linear finite elements employed in the formulation of Galerkin weighted-residual statements. The element-level linear shape functions are enhanced with supplementary polynomial bubble functions with undetermined coefficients. The best least squares minimization of the residual functional obtained from the insertion of these trial functions into model equations results in an algebraic system of equations whose solution provides the unknown coefficients in terms of element-level nodal values. The normal finite element procedures for the construction of stiffness matrices may then be followed with no extra degree of freedom incurred as a result of such enrichment. The performance of the proposed method has been tested on a number of benchmark linear transport equations with the results compared against the exact and standard linear element solutions. It has been observed that low order bubble enriched elements produce more accurate approximations than the standard linear elements with no extra computational cost despite employing relatively crude mesh. However, for the solution of strongly convection or reaction dominated problems significantly higher order enrichments as well as extra mesh refinements will be required.