Time-Continuous and Time-Discontinuous Space-Time Finite Elements for Advection-Diffusion Problems

TL;DR

This work provides a systematic evaluation of four space-time finite element discretizations—two time-continuous (C-PST, C-SST) and two time-discontinuous (D-PST, D-SST)—applied to advection-diffusion problems across parabolic and hyperbolic regimes. Through extensive space-time convergence studies, the authors establish robust quadratic spatial convergence and temporal convergence tending toward cubic order under appropriate boundary treatment and element choices, with D-PST achieving near-cubic temporal accuracy in several setups. The analyses reveal the critical role of time-dependent boundary conditions and the tensor-product versus simplex structure in determining convergence behavior, including an instance of “extra superconvergence” in nodal errors for advection-dominated cases. A highlight is the piston-ring heat-transfer demonstration using time-continuous simplex spacetime (C-SST), which showcases the method’s capability to handle complex topology changes and compute accurate heat-flux metrics in moving-boundary problems. Overall, the results provide practical guidance for selecting space-time discretizations and refinement strategies in advection-diffusion simulations and moving-domain applications, with implications for efficient parallel-in-time computation and heat-transfer analyses.

Abstract

We construct four variants of space-time finite element discretizations based on linear tensor-product and simplex-type finite elements. The resulting discretizations are continuous in space, and continuous or discontinuous in time. In a first test run, all four methods are applied to a linear scalar advection-diffusion model problem. Then, the convergence properties of the time-discontinuous space-time finite element discretizations are studied in numerical experiments. Advection velocity and diffusion coefficient are varied, such that the parabolic case of pure diffusion (heat equation), as well as, the hyperbolic case of pure advection (transport equation) are included in the study. For each model parameter set, the L2 error at the final time is computed for spatial and temporal element lengths ranging over several orders of magnitude to allow for an individual evaluation of the methods' spatial, temporal, and spacetime accuracy. In the parabolic case, particular attention is paid to the influence of time-dependent boundary conditions. Key findings include a spatial accuracy of second order and a temporal accuracy between second and third order. The temporal accuracy tends towards third order depending on how advection-dominated the test case is, on the choice of the specific discretization method, and on the time-(in)dependence and treatment of the boundary conditions. Additionally, the potential of time-continuous simplex space-time finite elements for heat flux computations is demonstrated with a piston ring pack test case.

Figures (24)

• Figure 1: Space-time discretization methods.
• Figure 2: Variants of simplex space-time discretization methods.
• Figure 3: Solution of IBVP 1 computed with four space-time discretization methods.
• Figure 4: Differences between $u$ and $u^h$.
• Figure 7: Organization of space-time convergence study based on $L_2$ error $e_{lm}$.