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A linear error on the one-dimensional Poisson coarse space is removed exactly

Example

Consider the homogeneous weak Dirichlet form for $- u^{''}$ on $(0, 1)$ , in the sense of Weak Dirichlet solutions for a divergence-form operator and Uniformly elliptic divergence-form operators and their sesquilinear forms, and the uniform mesh with four fine intervals. The continuous piecewise-linear nodal basis has interior hat functions $ϕ_{1}, ϕ_{2}, ϕ_{3}$ . Its stiffness matrix is

$A_{h} = 4 (\begin{array}{ccc} 2 & - 1 & 0 \\ - 1 & 2 & - 1 \\ 0 & - 1 & 2 \end{array}) .$

The coarser mesh has one interior node. Its piecewise-linear prolongation is $P = (1 / 2, 1, 1 / 2)^{T}$ . The vector $e = (1, 2, 1)^{T} = 2 P$ is represented exactly on the coarse space, so one exact Galerkin correction removes it.

This calculation uses the homogeneous weak-form branch, whose functional-space interface assumes Countable Choice. It establishes the finite matrix identity without asserting solution existence for a forcing or a finite-element error estimate, and it makes no claim about inhomogeneous trace data.

Facts & Assumptions

Given: The interval $(0, 1)$ , its zero-boundary weak Poisson form, the four-interval uniform mesh, and its continuous piecewise-linear nodal basis.

[F1]

Weak Dirichlet solutions for a divergence-form operator: in the homogeneous branch, a weak solution is an $H_{0}^{1}$ class satisfying $a (u, v) = F (v)$ for every test function; that branch assumes Countable Choice. Only the definition of the weak form is used here, not an existence theorem.

[F2]

Uniformly elliptic divergence-form operators and their sesquilinear forms: with coefficient $a^{11} = 1$ and $b = c = 0$ , the form is $a (u, v) = \int_{0}^{1} u^{'} (x) v^{'} (x) d x$ in the real case.

[F3]

Galerkin coarse matrices and the exact coarse-correction error map: $A_{H} = P^{T} A_{h} P$ and $C = I - P A_{H}^{- 1} P^{T} A_{h}$ when $A_{H}$ is invertible.

[F4]

Continuous piecewise-affine functions with zero endpoints lie in H1-zero: each continuous piecewise-affine function on $[0, 1]$ vanishing at $0$ and $1$ belongs to $H_{0}^{1} (0, 1)$ .

[A1]

Assume The Axiom of Countable Choice ( ${A C}_{ω}$ ) for the homogeneous Sobolev interface in [F1]. The finite matrix calculation uses this homogeneous branch.

Verification

1.1givenA1F1F2F4algebra

Each fine nodal hat is continuous piecewise affine and zero at both endpoints, so [F4] places it in $H_{0}^{1} (0, 1)$ , the test space in [F1] under [A1]. On each interval its derivative is constant. For $i = j$ , the integral $\int ϕ_{i}^{'} ϕ_{i}^{'}$ receives $1 / h$ from each of its two adjacent intervals, giving $2 / h = 8$ ; for adjacent indices, the shared interval contributes $- 1 / h = - 4$ ; nonadjacent supports have zero derivative-product integral. Thus the matrix of $a (\cdot, \cdot)$ is the displayed $A_{h}$ .

2.1F3step 1.1algebra

With $h = 1 / 4$ , direct multiplication gives $A_{h} P = (0, 4, 0)^{T}$ and $A_{H} = P^{T} A_{h} P = 4$ . Therefore [F3] gives $C = I - P (0, 1, 0)$ , which maps $(x_{1}, x_{2}, x_{3})^{T}$ to $(x_{1} - x_{2} / 2, 0, x_{3} - x_{2} / 2)^{T}$ .

3.1step 2.1algebra∎

For $e = (1, 2, 1)^{T}$ , step 2.1 yields $C e = (1 - 1, 0, 1 - 1)^{T} = 0$ . The error is exactly the prolongation of its single coarse value $2$ , so the exact coarse correction removes it.

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Sources

Susanne C. Brenner and L. Ridgway Scott, The Mathematical Theory of Finite Element Methods, 3rd ed., Chapter 6 (standard reference, not scraped)
Numerical Methods II, University of Heidelberg lecture notes, Chapter 10 (standard reference, not scraped)