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.
翻译:我们根据线性粒子产品和简单x型限制要素,构建了四种空间-时间限制元素离散变体。由此产生的离散体在空间中是连续连续的,在时间上是连续的。在第一次测试中,所有四种方法都适用于线性天平反动扩散模型问题。然后,在数字实验中研究时间不连续的时空-时间限制元素离散的趋同性特性。在数值实验中,对速度和传播系数存在差异,例如,在研究中包括纯扩散(热方程)的抛物体案例,以及纯倾斜(运输方程)的超双曲线案例。对于每个模型设定的参数,最后时间的L2误差是按空间和时间长度的长度来计算,以对方法的空间、时间、时间和时间的偏差进行单独评估。在参数上,对基于时间的边界条件的影响给予特别关注。关键结论包括第二顺序的空间精确度和第二和第三顺序之间的时间级间精确度(运输方位)的超常度案例,对具体时间的精确度-直径直径直度处理,对具体边界的精确度-直径直径直径直线-直径直线-直线-直径直度-直线性-直径直线性-直度-直度-直度-直度-直径直度-直度-直径直度-直径直度-直至。