A wavelet multiresolution interpolation Galerkin method for targeted local solution enrichment

Abstract

A novel wavelet multiresolution interpolation formula is developed for approximating continuous functions defined on an arbitrary two-dimensional domain represented by a set of scattered nodes. The present wavelet interpolant is created explicitly without the need for matrix inversion. It possesses the Kronecker delta function property and does not contain any ad-hoc parameters, leading to an excellent stability and usefulness for function approximation. Using the wavelet multiresolution interpolant to construct trial and weight functions, a wavelet multiresolution interpolation Galerkin method (WMIGM) is proposed for solving elasticity problems. In this WMIGM, the essential boundary conditions can be imposed with ease as in the conventional finite element method. The stiffness matrix can be efficiently obtained through semi-analytical integration using an underlying general database, instead of the numerical integration usually requiring a mesh. The accuracy of the WMIGM is examined through theoretical analysis and benchmark problems. Results demonstrate that the proposed WMIGM has an excellent accuracy, optimal rate of convergence and competitive efficiency, as well as an excellent stability against irregular nodal distribution. Most importantly, by adding more nodes into local region only, a high resolution of localized steep gradients can be achieved as desired without changing the existing nodes.