Abstract
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.
Highlights
Standard methods for the numerical solution of time-independent linear partial differential equations, such as finite difference, finite volume, and finite element methods, yield a sparse system of linear equations Au = b.Since memory requirement is often critical, an iterative solution method may be the only viable choice
One alternative line of research originates from Strang [15]. He proposed an algorithm for Toeplitz systems based on the conjugate gradient method with a circulant preconditioner: If M is a circulant approximation of A, the Fast Fourier Transform (FFT) can be used when applying M−1, and the resulting algorithm is memory lean and has a computational cost of near optimal order
Our inspiration comes from Neumann [13] and the research about linear integral equations and potential theory that Neumann
Summary
Standard methods for the numerical solution of time-independent linear partial differential equations, such as finite difference, finite volume, and finite element methods, yield a sparse system of linear equations. Since memory requirement is often critical, an iterative solution method may be the only viable choice. The iterative method is usually applied to a preconditioned problem
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