Abstract A Galerkin-based finite element recursion relation is used to solve the heat transport equation in two-dimensions. The finite element method (FEM) is a powerful technique that is commonly used for solving complex engineering problems. However, the implementation of the FEM in multi-dimensional problems can be computationally expensive. A finite element recursion algorithm based on bilinear triangular, bilinear quadrilateral and quadratic Lagrangian approximations are employed to discretize the 2-D advection-diffusion equation. This algorithm is an extension of the 1-D Chapeau (linear element) technique, which employed a tridiagonal recursion expression common to the classical central finite-difference approach. The global matrix is nine-diagonal (for 2-D) and is solved using a modified strongly implicit procedure and a left-to-right sweep method.

中文翻译：

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使用九对角求解器的输运方程的二维有限元递归关系

摘要 利用基于伽辽金的有限元递推关系求解二维传热方程。有限元法 (FEM) 是一种强大的技术，常用于解决复杂的工程问题。然而，在多维问题中实施 FEM 可能在计算上很昂贵。基于双线性三角形、双线性四边形和二次拉格朗日近似的有限元递归算法被用来离散化二维平流扩散方程。该算法是一维 Chapeau（线性元素）技术的扩展，它采用了经典中心有限差分方法常见的三对角递归表达式。