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An analytic box solution has exponential Galerkin p convergence

Example

Assume Countable Choice.

On the single reference square $Ω = (- 1, 1)^{2}$ let $u (x, y) = 1 / (4 - x)$ and use $a (v, w) = \int_{Ω} (\nabla v \cdot \nabla w + v w)$ on $H^{1} (Ω)$ with exact load $F (w) = a (u, w)$ . For every $p \geq 1$ , the tensor-degree- $p$ conforming Galerkin solution $u_{p}$ satisfies

∥ u - u_{p} ∥_{H^{1} (Ω)} \leq C \frac{1}{3} {(\frac{3}{4})}^{p}

for a fixed constant $C$ .

Facts & Assumptions

Given: Countable Choice, $p \geq 1$ , $u (x, y) = (4 - x)^{- 1}$ on $[- 1, 1]^{2}$ , and $H^{1}$ with norm $∥ v ∥_{H^{1}}^{2} = ∥ v ∥_{2}^{2} + ∥ \nabla v ∥_{2}^{2}$ .

[F1]

Analytic data converge exponentially under fixed-mesh p refinement: on the fixed box, reference derivative bounds $M ∣ ν ∣! R^{- ∣ ν ∣}$ with $R^{- 1} < q < 1$ yield a trace-matching polynomial with $H^{1}$ error at most $C_{R, q} M q^{p}$ .

[F2]

Cea transfers the fixed-mesh analytic p rate to the energy error: the exact coercive problem transfers this best approximation rate to its conforming Galerkin solution.

[F3]

Cauchy–Schwarz: $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥ ∥ y ∥$ , with equality exactly for dependent pairs: the inner-product Cauchy–Schwarz inequality bounds $∣ a (v, w) ∣$ by $∥ v ∥_{H^{1}} ∥ w ∥_{H^{1}}$ .

Verification

1.1F1givenalgebra

For a multi-index $ν = (n, j)$ , $D^{ν} u = 0$ when $j > 0$ , while $\partial_{x}^{n} u (x, y) = n! (4 - x)^{- n - 1}$ . Since $4 - x \geq 3$ on the square, $∣ D^{ν} u ∣ \leq \frac{1}{3} ∣ ν ∣! 2^{- ∣ ν ∣}$ in every case. Thus the hypothesis of [F1] holds with $M = 1 / 3$ , $R = 2$ , and $q = 3 / 4$ .

2.1F2F3step 1.1given∎

The form $a$ is the $H^{1}$ inner product. By [F3], $F (w) = a (u, w)$ is bounded, and $u$ is the exact variational solution by its definition. Applying [F2] with the constants established in step 1.1 gives $∥ u - u_{p} ∥_{H^{1}} \leq C (1 / 3) (3 / 4)^{p}$ .

Depends on

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Sources

Joachim Schöberl, Numerical Methods for Partial Differential Equations, p-version FEM (standard reference, not scraped)
Leszek F. Demkowicz, Mathematical Theory of Finite Elements (standard reference, not scraped)