THE STATE-OF-ART AND RECENT PROGRESSES OF THE REDUCED ELEMENT METHOD

The recently proposed reduced element method treats the highest-degree term in the conventional polynomial finite element (FE) solution as an error term, serving as a built-in maximum-norm error estimator. By taking the reduced solution (one degree lower) as the final solution, it simultaneously accomplishes problem solving and error evaluation without additional computations, thus achieving an adaptive FE algorithm that controls errors in the maximum norm. Initially developed for adaptive time-stepping algorithms in structural dynamic analysis, the reduced element was soon extended to two-dimensional boundary value problems (BVPs), where quadrilateral and triangular reduced elements, as well as space-time reduced elements, were constructed. Recently, preliminary progress has been made in three key areas: adaptive analysis using the dual error estimates (a priori plus a posteriori), the element-merging technique for space-time problems, and C¹ reduced elements for thin plate bending. This paper provides a comprehensive review of these research developments, briefly reports on the latest advancements in the three aspects and presents representative numerical examples to demonstrate the method's simplicity, effectiveness, generality, and reliability.