 Multigrid method

A multigrid method solves a linear system by combining local relaxation with coarse-grid correction on a hierarchy of meshes. The key is to divide the work by scale: a suitable relaxation removes rapidly oscillating approximation error, while coarse-grid correction addresses slowly varying approximation error that relaxation barely changes. A broad error hump occupies many fine-grid points but fewer coarse-grid points; it becomes easier to resolve by relaxation after enough coarsening. This complementary action is why multigrid methods are useful for large finite difference method and finite element method systems.

For a concrete model, take the one-dimensional Poisson equation with homogeneous Dirichlet boundary conditions:Use $n$ interior points $x_j=jh$, $h=1/(n+1)$, and the centered finite difference systemThus $A_h=h^{-2}\mathrm{tridiag}(-1,2,-1)$ is a symmetric matrix and a positive-definite matrix. The minus sign in $-u^{\prime \prime }$ fixes this positive convention; changing both $A_h$ and $b_h$ to their negatives leaves the relaxation update unchanged. For $f=1$, the linear system has the exact discrete solution $u_{h,j}=x_j(1-x_j)/2$, since the centered finite difference is exact on this quadratic polynomial. In general, the exact discrete solution need not equal the sampled continuous solution: solving the linear system removes algebraic approximation error, not the global discretization error.

Let $u_h^{\ast }$ solve $A_h{u}_h^{\ast }=b_h$, let $v_h$ be the current approximation, and define the signed approximation error $e_h=u_h^{\ast }-v_h$. The linear-system residual is $r_h=b_h-A_h{v}_h$, so $A_h{e}_h=r_h$. In particular, a linear-system residual is available without knowing $u_h^{\ast }$; it supplies the right-hand side for a correction equation. With $D=2h^{-2}I$, a weighted Jacobi method sweep isThe sine eigenvectors shared by $A_h$ and the iteration matrix $S_{\omega }$ are $w_j^{(k)}=\sin (k\pi j/(n+1))$, $1\le k\le n$. Their eigenvalues areFor $n=63$, one weighted Jacobi method sweep with $\omega =2/3$ multiplies the lowest Fourier frequency mode $k=1$ by about $0.999197$, but the highest Fourier frequency mode $k=63$ by about $-0.332530$. Four sweeps retain about $99.68\%$ of the first amplitude and $1.22\%$ of the second. The sign reversal of the second amplitude is harmless when its magnitude shrinks. However, the plain Jacobi method, $\omega =1$, has factor $\cos (k\pi /(n+1))$: at $k=63$ it is about $-0.998795$. A nearly alternating mode near the grid's Nyquist limit therefore persists almost as strongly as the smoothest mode. Plain Jacobi method convergence on a fixed Dirichlet boundary condition grid does not imply effective smoothing.^[ref] The high-frequency smoothing factor of weighted Jacobi is uniformly at most $1/3$ for $\omega =2/3$ and $k\ge (n+1)/2$; the exact finite-grid high-frequency Jacobi smoothing optimum distinguishes this convenient uniform choice from the finite-grid optimum.^[ref]

One two-grid cycle applies a few weighted Jacobi method sweeps, computes the linear-system residual, performs coarse-grid correction, and applies a few more sweeps. The first relaxation makes the remaining approximation error easier to represent on a coarse mesh; the last relaxation damps oscillatory components left after the transferred correction. In a geometric multigrid V-cycle, replace the expensive exact coarse solve by one recursive cycle on the coarse correction equation, starting its correction from zero. Repeat until a tiny coarsest linear system can be solved directly, then return through the levels by adding each interpolated correction and post-smoothing. The descent and return give the cycle its V shape.^[ref]

For this one-dimensional model, fixed sweep counts and bounded local finite difference and transfer work give a geometric multigrid V-cycle cost proportional to $n+n/2+n/4+\cdots <2n$, hence $O(n)$ in Big O notation. A fixed error reduction also needs a contraction factor bounded below one independently of $h$; per-cycle work alone does not establish that. Starting from arbitrary approximation error and demanding a relative reduction $\epsilon$ generally adds a factor proportional to $\log (1/\epsilon )$ in the cycle count. Full multigrid uses coarse solutions as initial guesses on successively finer meshes to target the global discretization error efficiently. The two-grid Poisson factor with weighted Jacobi illustrates favorable convergence for this model. Other partial differential equations, strong anisotropy, or poorly chosen coarse spaces and relaxation can defeat these choices; a linear-work guarantee for arbitrary linear systems does not follow.

A historical example of the multilevel viewpoint is Achi Brandt's April 1977 paper, which analyzes adaptive meshes and reports numerical experiments on linear and nonlinear boundary value problems. This is a documented development of the approach, not a claim that the paper was its first invention.^[ref] The figure above is generated from the displayed eigenvalues, with lines joining discrete nodes; it contains no experimental measurements.

The multigrid method is a computational technique used to solve a wide range of problems, particularly those involving partial differential equations (PDEs). It is designed to accelerate the convergence of iterative methods for solving such equations, especially when the problem is large and complex. ### Key Concepts: 1. **Multi-Level Approach**: The multigrid method works on multiple levels of discretization, typically on a hierarchy of grids with different resolutions.