An efficient solution algorithm for space-time finite element method

An efficient solution algorithm has been developed for space-time finite element method that is derived from time discontinuous Galerkin (TDG) formulation. The proposed algorithm features an iterative solver accelerated by a novel and efficient preconditioner. This preconditioner is constructed based on the block structure of coupled space-time system matrix, which is expressed as addition of Kronecker products of temporal and spatial submatrices. With this unique decomposition, the most computationally intensive operations in the iterative solver, i.e. matrix operations, are subsequently optimized and accelerated employing the inverse property of Kronecker product. Theoretical analysis and numerical examples both demonstrate that the proposed algorithm provides significantly better performance than the already developed implementations for TDG-based space-time FEM. It reduces the computational cost of solving space-time equations to the same order of solving stiffness equations associated with regular FEM, thereby enabling practical implementation of the space-time FEM for engineering applications.