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

TL;DR

This paper compares the computational complexity of space-time finite element formulations and time-marching schemes on h-refined grids near singularities, analyzing direct and iterative solvers through theoretical and numerical methods.

Contribution

It provides a comprehensive complexity analysis of space-time versus time-marching schemes, including direct and iterative solvers, with validation through numerical experiments.

Findings

Space-time formulations have different complexity profiles compared to time-marching schemes.

Iterative solvers show distinct performance characteristics on refined grids.

Numerical experiments confirm theoretical complexity estimates.

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 of comparing the computational time via time complexities of direct and iterative solvers. We focus on the space-time formulation with refined computational grids and on the corresponding time slabs, namely, computational grids obtained by taking the "cross-sections" of the refined space-time mesh. We compare the computational complexity of the space-time formulation and the corresponding time--marching scheme. Our consideration concerns the computational complexity of the multi-frontal solvers, the iterative solvers, as well as the static condensation. Numerical experiments with Octave confirm our theoretical findings.