We construct a monotone continuous $Q^1$ finite element method on the uniform mesh for the anisotropic diffusion problem with a diagonally dominant diffusion coefficient matrix. The monotonicity implies the discrete maximum principle. Convergence of the new scheme is rigorously proven. On quadrilateral meshes, the matrix coefficient conditions translate into specific a mesh constraint.

翻译：我们在均匀网格上为具有对角占优扩散系数矩阵的各向异性扩散问题构造了一个单调连续的 $Q^1$ 有限元方法。单调性蕴含了离散极大值原理。新格式的收敛性得到了严格证明。在四边形网格上，矩阵系数条件转化为特定的网格约束。