We consider the iterative solution of the discrete Poisson’s equation with Dirichlet boundary conditions. The discrete domain is embedded into an extended domain and the resulting system of linear equations is solved using a fixed point iteration combined with a multilevel circulant preconditioner. Our numerical results show that the rate of convergence is independent of the grid’s step sizes and of the number of spatial dimensions, despite the fact that the iteration operator is not bounded as the grid is refined. The embedding technique and the preconditioner is derived with inspiration from theory of boundary integral equations. The same theory is used to explain the behaviour of the preconditioned iterative method.

我们考虑具有狄利克雷边界条件的离散泊松方程的迭代解。离散域被嵌入到扩展域中，并且使用定点迭代结合多级循环预处理器来求解所得的线性方程组。我们的数值结果表明收敛速度与网格的步长和空间维数无关，尽管迭代算子在网格被细化时不受限制。嵌入技术和预处理器的灵感来自边界积分方程理论。相同的理论用于解释预处理迭代方法的行为。