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Uniform a priori estimates for elliptic problems with impedance boundary conditions

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  • We derive stability estimates in $ H^2 $ for elliptic problems with impedance boundary conditions that are uniform with respect to the impedance coefficient. Such estimates are of importance to establish sharp error estimates for finite element discretizations of contact impedance and high-frequency Helm-holtz problems. Though stability in $ H^2 $ is easily obtained by employing a ``bootstrap'' argument and well-established result for the corresponding Neumann problem, this strategy leads to a stability constant that increases with the impedance coefficient. Here, we propose alternative proofs to derive sharp and uniform stability constants for domains that are convex or smooth.

    Mathematics Subject Classification: Primary: 35J05; Secondary: 35B30.


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  • Figure 1.  Examples of domains with an obstacle satisfying Assumptions 1 and 2 with $ \Gamma_{{\rm{Diss}}} $ convex

    Figure 2.  Examples of domains with an obstacle satisfying Assumptions 1 and 2 with $ \Gamma_{{\rm{Diss}}} $ smooth

    Figure 3.  Examples of cavities satisfying Assumptions 1 and 2

    Figure 4.  Exceptional cavity satisfying Theorem 5.1

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