Multiresolution modeling with operator-customized wavelets derived from finite elements

In this article we describe the construction and application of operator-customized wavelets constructed out of general finite element interpolation functions (i.e., not just those of the Lagrange family). Unlike classical hierarchical bases, these wavelets are scale-orthogonal with respect to a given inner-product such as that arising from the weak-form of a differential operator. This property results in block-diagonal stiffness matrices that in turn permits the incremental computation of the solution in an efficient manner especially in the case of adaptive mesh refinement. As a specific example, we consider the design of such wavelets for the incremental and adaptive solution of fourth-order partial differential equations such as those governing the mechanics of thin elastic plates and shells.