A Fast Discontinuous Galerkin Finite Element Method for a Bond-Based Linear Peridynamic Model

Abstract

A fast discontinuous Galerkin finite element algorithm on a non-uniform mesh for solution of a one-dimensional bond-based linear peridynamic model with fractional noise is developed. It is based on the approximation of the stiffness matrix corresponding to the discontinuous Galerkin finite element method by its hierarchical representation. The fast algorithm reduces the storage requirement for the stiffness matrix from $\mathscr{O} (N^2 )$ to $\mathscr{O} (kN),$ where $k$ is a parameter controlling the accuracy of hierarchical matrices. The computational complexities of assembling the stiffness matrix and the Krylov subspace method for solving linear systems are also reduced from $\mathscr{O} (N^2)$ to $\mathscr{O} (kN).$ Numerical results show the utility of the numerical method.