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A coarse-grid correction improves a fine-grid approximation by solving a smaller equation for the part of its approximation error representable on a coarse mesh. Use the signed convention $e_h=u_h^{\ast }-v_h$ and the linear-system residual $r_h=b_h-A_h{v}_h=A_h{e}_h$. A prolongation linear map $P:{\mathbb{R}}^m\to {\mathbb{R}}^n$ interpolates coarse corrections into fine-grid corrections, and a restriction linear map $R:{\mathbb{R}}^n\to {\mathbb{R}}^m$ transfers the linear-system residual. Solve $A_H{z}_H=R{r}_h$ exactly or approximately and update $v_h\leftarrow v_h+P{z}_H$. The plus sign follows from defining the approximation error as exact minus approximate. It is the correction equation's right-hand side $R{r}_h$, rather than merely restricted solution values, that determines the update.

For the multigrid method model $A_h=h^{-2}\mathrm{tridiag}(-1,2,-1)$, take $n=2m+1$, so $H=2h$ and the coarse nodes coincide with every second fine node. Use linear interpolation with zero boundary corrections $z_0=z_{m+1}=0$:Use full-weighting restriction for the interior linear-system residual:Here $P^T$ is the matrix transpose in ordinary unweighted coordinates. With these specific transfers, the Galerkin coarse-grid operator isThus it equals the rediscretized Poisson equation operator on this constant-coefficient one-dimensional mesh. The equality is special to the stated model and transfers. In particular, dropping $h^{-2}$ from the fine matrix while using $H^{-2}$ on the coarse mesh changes the correction unless the right-hand sides and transfers are rescaled consistently.

Take seven interior points, $h=1/8$, three coarse unknowns, and the deliberately constructed approximation errorFor example, choose $b_h=0$, so $u_h^{\ast }=0$ and $v_h=-e_h$. Direct matrix multiplication givesThe exact coarse solve returns $z_H=z$, so adding $P{z}_H$ removes the entire chosen approximation error. A broad triangular error can have a sharply localized linear-system residual: the latter measures discrete curvature, not error height. This is why the two should not be confused. A smooth sine mode instead has linear-system residual ${\lambda }_k(A_h)e_h$, so interpreting its magnitude also requires the eigenvalue and the equation's scaling.

The algebra explains the exact cancellation beyond this example. Assume $P$ has linearly independent columns and choose an invertible Galerkin coarse-grid operator $A_H=R{A}_{h}P$. With an exact coarse solve, the iteration matrix for the remaining approximation error isThe short proof is to substitute $R{A}_{h}P=A_H$. Thus every vector in the column space of $P$ is eliminated, and the corrected linear-system residual restricts to zero. These are exact projection identities, not a statement that every visually smooth vector belongs to that column space. If $A_h$ is a symmetric matrix and a positive-definite matrix, and $R=c{P}^T$ with $c>0$, then $A_H$ is also a positive-definite matrix. Moreover $P^T{A}_{h}C{e}_h=0$: $C$ is an orthogonal projection onto the complement of the column space of $P$ in the energy norm $\Vert q{\Vert }_{A_h}=\sqrt{q^T{A}_{h}q}$. Equivalently, the added correction is the best approximation to $e_h$ from that column space in this energy norm. Without these symmetry and transfer assumptions, the projection need not be orthogonal.^[ref]

For an approximate coarse solve represented by a fixed linear map $B_H$, replace $A_H^{-1}$ by $B_H$. ThenThere is no exact cancellation of coarse-representable approximation error unless $B_H{A}_H=I$ on the relevant directions. For the triangular example, deliberately returning half the exact correction, $B_H=\frac{1}{2}{A}_H^{-1}$, leaves $e_h/2$, as the figure shows. This controlled mathematical comparison illustrates an incomplete coarse solve; it is not a measured performance result or a prescription for a geometric multigrid V-cycle. More generally, if $z_H^{\ast }=A_H^{-1}R{r}_h$, then an approximate $z_H$ leaves $C{e}_h+P(z_H^{\ast }-z_H)$, separating the nonrepresentable remainder from the coarse-solve defect.

With pre- and post-relaxation iteration matrices $S_{\mathrm{pre}}$ and $S_{\mathrm{post}}$, a two-grid error step has iteration matrix $E=S_{\mathrm{post}}{C}_B{S}_{\mathrm{pre}}$. A geometric multigrid V-cycle supplies $B_H$ by one recursive coarse cycle from zero, with a direct solve only at the bottom. Effective smoothing, a coarse column space that approximates the surviving approximation error, and a sufficiently accurate coarse correction must work together. Coarsening alone supplies no convergence guarantee, and correcting an arbitrary fine-grid vector exactly would require a coarse column space large enough to contain it.

Full multigrid solves first on a coarse mesh, interpolates that approximation to the next finer mesh, and applies a fixed number of geometric multigrid V-cycles before proceeding to the next level. If interpolation introduces only the expected discretization-scale error and the cycle count reduces the inherited error sufficiently at every level, the final algebraic error matches the fine-grid discretization error. The geometric sum of level costs is then $O(N)$, unlike reducing an arbitrary fine-grid initial error by an arbitrarily small tolerance.

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.