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

We construct four variants of space-time finite element discretizations based on linear tensor-product and simplex-type elements. The resulting are continuous in space, or discontinuous time. In a first test run, all methods applied to scalar advection-diffusion model problem. Then, the convergence properties time-discontinuous studied numerical experiments. Advection velocity diffusion coefficient varied, such that parabolic case pure (heat equation), as well as, hyperbolic advection (transport equation) included study. For each parameter set, L 2 $$ {L}_2 error at final time is computed for spatial temporal lengths ranging over several orders magnitude allow an individual evaluation methods' spatial, temporal, accuracy. case, particular attention paid influence time-dependent boundary conditions. Key findings include accuracy second order between third order. tends toward depending how advection-dominated is, choice specific discretization method, time-(in)dependence treatment Additionally, potential time-continuous simplex elements heat flux computations demonstrated with piston ring pack subtractive manufacturing case.