A monolithic space-time temporal multirate finite element framework for interface and volume coupled problems

TL;DR

The paper tackles solving coupled multiphysics problems with different temporal scales by a monolithic space-time multirate framework that uses a tensor-product Galerkin finite element method on time-slabs $S_m=\Omega\times I_m$. Subproblems are allowed to employ distinct temporal meshes while coupling is enforced within a single variational formulation, yielding fully implicit, subcycling-free solutions. A rigorous abstract formulation with block structure $A_1,A_2,B_1,B_2$ and compatible function spaces is paired with nonmatching-time integration via a restriction matrix $R$, enabling accurate assembly on space-time slabs. Five numerical tests from 1+1D to 3+1D Mandel/footing demonstrate robustness, convergence, and efficiency gains, validating the approach for interface-coupled heat–wave and volume-coupled poroelasticity problems.

Abstract

In this work, we propose and computationally investigate a monolithic space-time multirate scheme for coupled problems. The novelty lies in the monolithic formulation of the multirate approach as this requires a careful design of the functional framework, corresponding discretization, and implementation. Our method of choice is a tensor-product Galerkin space-time discretization. The developments are carried out for both prototype interface- and volume coupled problems such as coupled wave-heat-problems and a displacement equation coupled to Darcy flow in a poro-elastic medium. The latter is applied to the well-known Mandel's benchmark and a three-dimensional footing problem. Detailed computational investigations and convergence analyses give evidence that our monolithic multirate framework performs well.

Figures (18)

• Figure 1: Example for possible temporal meshes $\mathcal{T}_k^1, \mathcal{T}_k^2, \mathcal{T}_k^{\text{fine}}$ and $\mathcal{T}_k^{\text{coarse}}$
• Figure 2: We show a snapshot of a block matrix which is solved in every time-step of the coarse mesh $\mathcal{T}_k^{\text{coarse}}$. The off-diagonal entries correspond to the coupling conditions. We emphasize that this sketch formally represents both interface-coupling and volume-coupling. In interface-coupling, $B_1$ and $B_2$ represent cross terms associated to the interface trial and test functions, while in volume-coupling, domain cross terms are contained in $B_1$ and $B_2$. Specific examples of $B_1$ and $B_2$ are provided in Section \ref{['sec_model_problems']}.
• Figure 3: Example of $\mathop{\mathrm{dG}}\limits(1)$ basis functions on the temporal meshes $\mathcal{T}_k^1$ and $\mathcal{T}_k^2$
• Figure 4: Analytical solution for the 1+1D heat and wave equation problem (solid in upper half and fluid in lower half of the images)
• Figure 5: Convergence plots for the 1+1D heat and wave equation problem with a finer fluid temporal mesh and uniform refinement in space and time

Theorems & Definitions (5)

• Remark 2.1
• Remark 2.2
• Remark 2.3
• Remark 2.4
• Remark 2.5