Time‐continuous and time‐discontinuous space‐time finite elements for advection‐diffusion problems

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$$ {L}_2 $$ 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 space‐time 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 toward 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 and a subtractive manufacturing test case.

Model order reduction for deforming domain problems in a time‐continuous space‐time setting

“…A systematic comparison of time-continuous and time-discontinuous space-time formulations has been performed for advection-diffusion problems. 1 However, as we would like to stress again, the time-continuous description is essential to directly apply established MOR techniques to deforming domain problems later.…”

Arbitrary mesh‐moving velocity‐based space‐time finite element method for large deformation analysis of solids

“…It is worth noting that the discretization described here can be categorized into the Time-discontinuous prismatic space-time method according to von Danwitz et al 52 For constructing a system of equations for 2-D problems, (28a) is reshaped as…”

“…Figure 14 plots the variations in radial-stress (𝜎 rr ) and hoop-stress (𝜎 𝜃𝜃 ) with the radial distance in the hypoelastic cylinder along with the exact solution (Equation 52). It is worth mentioning that in the case of hyperelastic material, analytical solutions, such as Equation (52), cannot be obtained. Therefore, in Figure 15, the computed stress is plotted along with the stress at the initial stage (𝜙 = 0).…”