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Well-definedness and best approximation of the Maxwell Ritz map

Statement

In the Countable Choice branch of Maxwell curl-energy Ritz projection, the equation defining $R_{h} u$ has a unique solution for each $u \in V$ . Moreover, with $∥ v ∥_{a_{+}}^{2} = a_{+} (v, v)$ , $∥ u - R_{h} u ∥_{a_{+}} \leq \inf_{v_{h} \in V_{h}} ∥ u - v_{h} ∥_{a_{+}} .$

Facts & Assumptions

Given: The finite-dimensional real edge space and symmetric form $a_{+}$ from Maxwell curl-energy Ritz projection, with uniformly positive definite $ϵ$ and $μ^{- 1}$ .

[A1]

Countable Choice is inherited from the H(div) and affine Piola definitions that supply the FE spaces; the local linear-algebra argument below uses only the given finite space. (The Axiom of Countable Choice ( ${A C}_{ω}$ ), Maxwell curl-energy Ritz projection)

[F1]

The mass form $m$ and curl form $a$ are symmetric, and their sum satisfies $a_{+} (v, v) \geq c (∥ v ∥_{L^{2}}^{2} + ∥ curl v ∥_{L^{2}}^{2})$ for some $c > 0$ . (Maxwell curl-energy Ritz projection)

Proof

1.1A1F1givenalgebra

By [F1], the Gram matrix of $a_{+}$ in any basis of the finite-dimensional $V_{h}$ is symmetric positive definite. Hence its kernel is zero and it is invertible, so the linear system defining $R_{h} u$ has exactly one solution.

1.2F1givenalgebra

Subtracting the defining equation from itself gives $a_{+} (u - R_{h} u, v_{h}) = 0$ for every $v_{h} \in V_{h}$ .

2.1A1F1step 1.2algebra∎

For arbitrary $w_{h} \in V_{h}$ , write $u - w_{h} = (u - R_{h} u) + (R_{h} u - w_{h})$ . The second summand lies in $V_{h}$ , so [step 1.2] makes the cross term zero in the $a_{+}$ inner product. Thus $∥ u - w_{h} ∥_{a_{+}}^{2} = ∥ u - R_{h} u ∥_{a_{+}}^{2} + ∥ R_{h} u - w_{h} ∥_{a_{+}}^{2} \geq ∥ u - R_{h} u ∥_{a_{+}}^{2}$ . Taking the infimum proves the claim.

Depends on

Used by

Smooth PEC edge finite element Maxwell error from compatible projections Theorem

Cited to discharge well-definedness by Maxwell curl-energy Ritz projection.

Dependency tree · two levels

8 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Long Chen, Finite Element Methods for Maxwell Equations, complete lecture notes (standard reference, not scraped)