High-dimensional transport equations frequently occur in science and engineering. Computing their numerical solution, however, is challenging due to its high dimensionality. In this work we develop an algorithm to efficiently solve the transport equation in moderately complex geometrical domains using a Galerkin method stabilized by streamline diffusion. The ansatz spaces are a tensor product of a sparse grid in space and discontinuous piecewise polynomials in time. Here, the sparse grid is constructed upon nested multilevel finite element spaces to provide geometric flexibility. This results in an implicit time-stepping scheme which we prove to be stable and convergent. If the solution has additional mixed regularity, the convergence of a $2d$-dimensional problem equals that of a $d$-dimensional one up to logarithmic factors. For the implementation, we rely on the representation of sparse grids as a sum of anisotropic full grid spaces. This enables us to store the functions and to carry out the computations on a sequence regular full grids exploiting the tensor product structure of the ansatz spaces. In this way existing finite element libraries and GPU acceleration can be used. The combination technique is used as a preconditioner for an iterative scheme to solve the transport equation on the sequence of time strips. Numerical tests show that the method works well for problems in up to six dimensions. Finally, the method is also used as a building block to solve nonlinear Vlasov-Poisson equations.

翻译：高维输运方程在科学与工程领域频繁出现，但其高维特性使得数值求解极具挑战性。本文基于流线扩散稳定的Galerkin方法，提出了一种能高效求解中等复杂几何区域内输运方程的算法。解空间由空间域稀疏网格与时间域间断分片多项式的张量积构成，其中稀疏网格建立在嵌套多层次有限元空间上以保证几何灵活性。由此得到隐式时间推进格式，我们证明了其稳定性和收敛性。当解具有额外混合正则性时，$2d$维问题的收敛性在对数因子级别等同于$d$维问题。实现过程中，我们将稀疏网格表示为各向异性全网格空间的叠加，从而利用解空间的张量积结构，在正则全网格序列上完成函数存储与计算。这种方法可兼容现有有限元库及GPU加速。我们采用组合技术作为迭代格式的预处理器，在时间条序列上求解输运方程。数值实验表明该方法在高达六维的问题中表现良好。最后，该算法还被用作构建模块求解非线性Vlasov-Poisson方程。