An hp-Galerkin method with fast solution for linear peridynamic models in one dimension

The computational work and memory requirement are bottlenecks for Galerkin finite element methods for peridynamic models because of their non-locality. In this paper, fast Galerkin and hp-Galerkin finite element methods are introduced and analyzed to solve a steady-state peridynamic model. We present a fast solution technique to accelerate non-square Toeplitz matrix-vector multiplications arising from piecewise-linear, piecewise-quadratic and piecewise-cubic Galerkin methods. This fast solution technique is based on a fast Fourier transform and depends on the special structure of coefficient matrices, and it helps to reduce the computational work from O(N3) required by traditional methods to O(Nlog2N) and the memory requirement from O(N2) to O(N) without using any lossy compression, where N is the number of unknowns. The peridynamic model admits solutions having jump discontinuities. For problems with discontinuous solutions, we therefore introduce a piecewise-constant Galerkin method and give an h- and p-refinement algorithm. Then, we develop fast hp-Galerkin methods based on hybrid piecewise-constant/piecewise-linear and piecewise-constant/piecewise-quadratic finite element approximations. The new method reduces the computational work from O(N3) required by the traditional methods to O(Nlog2N) and the memory requirement from O(N2)to O(N).