Elliptical PDEs are at the core of many computational problems. Sometimes it is necessary to solve them on fine meshes, which entail huge memory footprints and low computational speeds. We provide a method to solve elliptical PDEs on fine grids with lowered memory consumption and improved convergence.

This paper considers one typical elliptical PDE – the Poisson equation on various polygonal domains with Dirichlet boundary conditions. The Finite Element Method (FEM) is used for numerical solution. FEM approximates a two-dimensional PDE as a system of linear equations $Au=f$. For an n-by-n mesh grid the sparse representation of A has the size of $O(n^2)$. We replace the sparse matrix representation with a Quantized Tensor Train (QTT) representation to obtain $O({(\log n)}^{\alpha })$ time and memory complexity to construct both A and f. This is ensured by constant-bounded QTT approximation ranks. AMEn solver is used on the final linear system, and its iterations are faster for low rank QTT approximation. To avoid rank growth caused by the intrinsic structure of A we introduce a new operation z-kron which constructs a matrix with rows and columns permuted into so called z-order. An algorithm to construct A in z-order directly in QTT with $O(1)$ ranks w.r.t. n is provided.

The proposed method is used to solve the Poisson equation on two different polygonal domains. Experiments show that our approach significantly improves memory consumption and speed over classical sparse-matrix based partial differential equation solvers like FEniCS.

椭圆形PDE是许多计算问题的核心。有时有必要在精细的网格上求解它们，这会占用大量内存并降低计算速度。我们提供了一种方法来解决细网格上的椭圆PDE，从而降低了内存消耗并提高了收敛性。

本文考虑一种典型的椭圆PDE-具有Dirichlet边界条件的各种多边形区域上的Poisson方程。有限元方法（FEM）用于数值求解。有限元法将二维PDE近似为线性方程组$\mathrm{一种}ü=F$。对于n × n的网格，A的稀疏表示的大小为$Ø（{ñ}^2）$。我们将量化矩阵张量（QTT）替换为稀疏矩阵表示，以获得$Ø（{（\mathrm{日志}ñ）}^{\alpha }）$构造A和f的时间和内存复杂度。这由恒定范围的QTT近似秩来确保。AMEn求解器用于最终的线性系统，对于低秩QTT逼近，其迭代速度更快。为了避免由A的固有结构引起的秩增长，我们引入了一个新的操作z-kron，该操作构造了一个行和列排列成所谓的z阶的矩阵。直接在QTT中以z顺序构造A的算法$Ø（\mathrm{1个}）$提供了n个等级。

该方法用于求解两个不同多边形区域上的泊松方程。实验表明，与传统的基于稀疏矩阵的偏微分方程求解器（例如FEniCS）相比，我们的方法显着提高了内存消耗和速度。