The rise of automation in industrial production has highlighted the importance of non-contact joining methods like laser beam welding. Lasers are ideal for joining metals due to their precision, short cycle times, and minimal heat-affected zones. However, solidification cracks often occur, particularly with materials that have a wide melting range. These cracks result from the high cooling rates in laser welding, which oversaturate the residual melt with alloy elements. To improve and optimize the laser beam welding process, it is crucial to develop a quantitative understanding of the mechanisms behind the initiation of solidification cracks and their relationship to key process parameters. This knowledge will enable the refinement of techniques to mitigate these defects and enhance the reliability of laser-welded joints. As a step toward this goal, we here present large time-dynamic and HPC-supported finite element simulation of the mechanical and thermal stresses of thermoelastic and thermoplastic laser beam welding problems using realistic process parameters and a high resolution. To obtain efficient and highly parallel scalable simulations, a two-level overlapping Schwarz method with GDSW (Generalized Dryja–Smith–Widlund) and RGDSW (Restricted Generalized Dryja–Smith–Widlund) coarse space is used as a preconditioner to acccelerate the convergence of the Generalized Minimal Residual Method (GMRES). The parallel implementation of these numerical solver strategies is done, exploiting the software libraries FE2TI and PETSc, using a hybrid OpenMP/MPI approach. Scientific support and HPC resources are provided by the Erlangen National High Performance Computing Center (NHR@FAU) of the Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU) under the NHR project k109be10.