A Performance Comparison of Adaptive Operator-Customized Wavelet Basis and Adaptive -Refinement Methods for 2-D Finite-Element Analysis

Abstract

This paper compares the performance of the popular adaptive $h$ -refinement ( $h\text{R}$ ) technique for the finite-element method (FEM) with the adaptive version of the recently presented operator-customized wavelet basis (OCWB). This new method is a combination of the second-generation wavelet theory with hierarchical basis, which is a multi-resolution basis, and, when applied to the FEM, the solution is split into different levels of detail. These levels, also referred to as scales, are composed by compactly supported functions, allowing the detail to be added only on the chosen regions. Although this strategy reduces the total system dimension with sufficiently small error, all the scales are naturally coupled, which means that there are interactions between functions composing different detail levels. This coupling forces the whole multi-resolution system to be recomputed when a detail is added, a kind of redundancy that also occurs on the $h\text{R}$ method. Using the second-generation wavelet theory, functions are custom-designed to decouple the system, which means the interactions between the functions of different levels are eliminated, and it is possible to solve for further detail independently of previous scales. This property significantly increases the performance of the algorithm. Conversely, the formulation—and, consequently, the algorithm design—complexity is also increased, which is the reason why there are such few applications on the subject. Like the $h\text{R}$ method, an adaptive OCWB can be programmed with various strategies; since the results are shown in terms of processing time on a regular PC, both the algorithms have been developed with similar structures.