Complexity of direct and iterative solvers on space–time formulations and time-marching schemes for h-refined grids towards singularities

Abstract

We study computational complexity aspects for Finite Element formulations considering hypercubic space–time full and time-marching discretization schemes for h-refined grids towards singularities. We perform a relatively comprehensive study comparing the computational time via time complexities of direct and iterative solvers. We focus on the space–time formulation with refined computational grids and the corresponding time slabs, namely, computational grids obtained by taking the “cross-sections” of the refined space–time mesh. We estimate the computational complexity of solving systems of linear equations for multidimensional meshes with arbitrary dimensional singularities encountered in space–time formulation and time-marching schemes. The choice between space–time formulation and time-marching schemes depends entirely on the problem’s nature and the properties that need to be addressed. Thus, the paper aims to discuss the computational complexities of both approaches rather than suggest a better formulation. Our considerations concern the computational complexity of sequental execution of the multi-frontal solvers, the iterative solvers, and the static condensation. Numerical experiments with Octave confirm our theoretical findings.