By Stefan Kurz, Bernhard Auchmann (auth.), Ulrich Langer, Martin Schanz, Olaf Steinbach, Wolfgang L. Wendland (eds.)
This quantity includes 8 cutting-edge contributions on mathematical features and functions of quickly boundary point equipment in engineering and undefined. This covers the research and numerics of boundary fundamental equations through the use of differential varieties, preconditioning of hp boundary aspect equipment, the applying of quickly boundary point tools for fixing hard difficulties in magnetostatics, the simulation of micro electro mechanical structures, and for touch difficulties in sturdy mechanics. different contributions are on fresh effects on boundary point tools for the answer of temporary problems.
This e-book is addressed to researchers, graduate scholars and practitioners engaged on and utilizing boundary aspect tools. All contributions additionally express the good achievements of interdisciplinary study among mathematicians and engineers, with direct purposes in engineering and industry.
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Example text
1. For the vectorial case n = 3, p = 1, s = −1 see [18, 19]. (48) 32 S. Kurz and B. Auchmann Consider the surface current density k ∈ H−1/2 (divΓ , ∂ Ω ) and the surface charge density σ ∈ H −1/2 (∂ Ω ). For p = 1, (k, σ ) can be defined as vector and scalar proxies of the differential forms (γ , ϕ ), respectively. The first equation in (48) then translates into divΓ k + iωσ = 0, the continuity equation, where ω is the angular frequency. Therefore, (48) is called generalized continuity equation.
Auchmann with the norm |ωˆ | 1/2 H⊥ Λ p (Γ ) = | ∗ ωˆ | H 1/2 q−1 Λ (Γ ) . Note that the maps ±1/2 p ±1/2 q−1 Λ (Γ ), ±1/2 : H⊥ Λ p (Γ ) ±1/2 q−1 (Γ ) ∗:H ∗ Λ (Γ ) → H⊥ →H Λ are isometric isomorphisms, and the following inclusions hold [36, Thm. 2], H 1/2 p −1/2 p 1/2 −1/2 p Λ (Γ ) ⊂ L2 Λ p (Γ ) ⊂ H Λ (Γ ), H⊥ Λ p (Γ ) ⊂ L2 Λ p (Γ ) ⊂ H⊥ Λ (Γ ). Remark 3. 1. For smooth boundaries, the spaces H 1/2 p Λ (Γ ) 1/2 and H⊥ Λ p (Γ ) are identical. 1/2 1/2 2. A fully intrinsic characterization of the spaces H Λ p (Γ ) and H⊥ Λ p (Γ ) is available in the case of polyhedral domains for n = 3, p = 1.
The last equation follows with (8) and (15). Corollary 2. The norms of traces of the Green kernel on the far boundary are bounded by the norms of traces of the scalar fundamental solution, |γDR G p |XL2 (Γ R ) ≤ |γDR G0 |XL2 (Γ R ) , |(γNR − ikγDR )G p |XL2 (Γ R ) ≤ |(γNR − ikγDR )G0 |XL2 (Γ R ) + |dγDRG0 |XL2 (Γ R ) . (59a) (59b) The inequality for the Neumann trace features an additional term, which involves tangential derivatives. Differential Forms and Boundary Integral Equations for Maxwell-Type Problems 35 Proof.