Generalized hierarchical bases:: A wavelet-Ritz-Galerkin framework for Lagrangian FEM

Purpose - This paper presents a novel framework for solving elliptic partial differential equations (PDEs) over irregularly spaced meshes on bounded domains. Design/methodology/approach - Second-generation wavelet construction gives rise to a powerful generalization of the traditional hierarchical basis (HB) finite element method (FEM). A framework based on piecewise polynomial Lagrangian multiwavelets is used to generate customized multiresolution bases that have not only HB properties but also additional qualities. Findings - For the 1D Poisson problem, we propose - for any given. order of approximation - a compact closed-form wavelet basis that block-diagonalizes the stiffness matrix. With this wavelet choice, all coupling between the coarse scale and detail scales in the matrix is eliminated. In contrast, traditional higher-order (n > 1) HB do not exhibit this property. We also achieve full scale-decoupling for the 2D Poisson problem on an irregular mesh. No traditional HB has this quality in 2D. Research limitations/implications - Similar techniques may be applied to scale-decouple the multiresolution finite element (FE) matrices associated with more general elliptic PDES. Practical implications - By decoupling scales in the FE matrix, the wavelet formulation lends itself particularly well to adaptive refinement schemes. Originality/value - The paper explains second-generation wavelet construction in a Lagrangian FE context. For ID higher-order and 2D first-order bases, we propose a particular choice of wavelet, customized to the Poisson problem. The approach generalizes to other elliptic PDE problems.