One approach with rising popularity in analyzing time-dependent problems in science and engineering is the so-called space-time finite-element method that utilized finiteelements in both space and time. A common ansatz in this context is to divide the mesh in temporal direction into so-called space-time slabs, which are subsequently weakly connected in time with a Discontinuous Galerkin approach. The corresponding jumpterm, which is responsible for imposing the weak continuity across space-time slabs can be challenging to compute, in particular in the context of deforming domains. Ensuring a conforming discretization of the space-time slab at the top and bottom in time direction simplifies the handling of this term immensely. Otherwise, a computationally expensive and error prone projection of the solution from one time-level to another is necessary. However, when it comes to simulations with deformable domains, e.g. for free-surface flows, ensuring conforming meshes is quite laborious. A possible solution to this challenge is to extrude a spatial mesh in time at each time-step resulting in the so-called time-discontinuous prismatic space-time (D-PST) method. However, this procedure is restricted to finite-elements of 1st order in time. We present a novel algorithmic approach for arbitrarily discretized meshes by flipping the mesh in time-direction for each time-step. This ansatz allows for a simple evaluation of the jump-term as the mesh is always conforming. The cost of flipping the mesh around its symmetry plane in time scales with the number of nodes, which makes it computationally cheaper than an additional update of the mesh to enforce conformity or the evaluation of a projection. We validate the approach on various physical problems with and without deforming domains.
翻译:在分析科学和工程中基于时间的问题方面越来越受欢迎的一种方法是所谓的时空定点方法,这种方法在时空上和时空上使用定点。一种常见的反射法是将时空方向的网格分为所谓的时空平板,随后与不连续的Galerkin方法在时间上连接较弱。对应的跳跃期是造成时空平板之间连续性薄弱的原因之一,这种在时空平板之间造成不稳定的连续性,特别是在变形域方面。确保时空平板之间的连续性较弱的跳跃期可能是难以计算的方法。在时空平时平时平时使时平时平时平时的平时平时平时平时平时平时平时平时,平时平时平时平时平时平时平时平时平时平时平时平时平时平时,平时平时平时平时平时平时平时平时平时平时,平时平时平时平时平时平时平时平时平时平时,平时平时平时平时平时平时平时平时,平时平时平时平时平时平时,平时平时平时平时平时平时平时平时平时平时,平时平时平时平时平时平时平时平时平时平时平时平时,平时平时平时平时平时平时平时平时,平时平时平时平时,平时平时平时平时平时平时平时平时平时,平时平时平时平时平时,平时,平时平时平时,平时平时平时平时平时,平时平时平时平时平时平时,平时平时平时平时平时平时,平时,平时平时,平时,平时,平时平时平时平时平时平时平时平时平时平时平时平时平时平时平时平时平时平时平时平时平时平时平时平时,平时,平时平时,平时平时平时平时平时平时平时平时平时平时,平时,平时平时