Abstract

Logarithmically optimal in theory and fast in practice, direct algorithms for implementing a tensor product finite element method (FEM) based on tensor products of 1D high-order FEM spaces on multi-dimensional rectangular parallelepipeds are proposed for solving the N-dimensional Poisson-type equation \( - \Delta u + \alpha u = f\) (\(N \geqslant 2\)) with Dirichlet boundary conditions. The algorithms are based on well-known Fourier approaches. The key new points are a detailed description of the eigenpairs of the 1D eigenvalue problems for the high-order FEM, as well as fast direct and inverse eigenvector expansion algorithms that simultaneously employ several versions of the fast Fourier transform. Results of numerical experiments in the 2D and 3D cases are presented. The algorithms can be used in numerous applications, in particular, to implement tensor product high-order finite element methods for various time-dependent partial differential equations, including the multidimensional heat, wave, and Schrödinger ones.

中文翻译：

---

泊松型方程张量积高阶有限元的快速傅立叶求解器

摘要

在理论上对数最优并且在实践中很快，提出了一种基于多维一维平行六面体上的一维高阶有限元空间张量积实现张量积有限元方法（FEM）的直接算法，用于求解N维泊松型等式\（-\ Delta u + \ alpha u = f \）（\（N \ geqslant 2 \））与Dirichlet边界条件。该算法基于众所周知的傅里叶方法。关键的新点是对高阶FEM的一维特征值问题的特征对的详细描述，以及同时采用快速傅里叶变换的几种版本的快速直接和逆特征向量展开算法。给出了在2D和3D情况下的数值实验结果。该算法可用于众多应用中，尤其是可针对各种与时间相关的偏微分方程（包括多维热，波动和薛定ding方程）实施张量积高阶有限元方法。