Cost of space-time formulations: a study on the performance of direct and iterative solvers on space-time formulations versus time-marching schemes

We focus on finite element method computations for time-dependent problems. We prove that the computational cost of the space-time formulation is higher than the cost of the time-marching schemes. This applies to both direct and iterative solvers. It concerns both uniform and adaptive grids. The only exception from this rule is the h adaptive space-time simulation of the traveling point object, resulting in refinements towards their trajectory in the space-time domain. However, if this object has wings and the mesh refinements capture the shape of the wing (if the mesh refinements capture any two-dimensional manifold) the space-time formulation is more expensive than time-marching schemes. We also show that the cost of static condensation for the higher-order finite element method with hierarchical basis functions is always higher for space-time formulations. Numerical experiments with Octave confirm our theoretical findings.