By Jens M. Melenk
Many partial differential equations bobbing up in perform are parameter-dependent difficulties which are of singularly perturbed variety. sought after examples contain plate and shell versions for small thickness in good mechanics, convection-diffusion difficulties in fluid mechanics, and equations coming up in semi-conductor gadget modelling. universal positive factors of those difficulties are layers and, in terms of non-smooth geometries, nook singularities. Mesh layout rules for the effective approximation of either positive factors by means of the hp-version of the finite aspect process (hp-FEM) are proposed during this quantity. For a category of singularly perturbed difficulties on polygonal domain names, powerful exponential convergence of the hp-FEM in keeping with those mesh layout rules is proven carefully.
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Extra info for hp-Finite Element Methods for Singular Perturbations
Example text
17) that is suited for polynomial approximation results. Although we apply our regularity results to high order approximations, they could also be employed to rigorously establish these low order approximation results. We present such an approach in Sec. 6. While a fair body of literature is available for lower order methods on meshes that can capture the boundary layer behavior of the solution, the situation is not so well developed for spectral and hp-methods. The first result in one dimension appears to be [113,114].
8, we construct a specific approximant to uε that leads to an upper bound in the last infimum. 8. 4. 3). 5) satisfies u − uN L2 (Ω) + ε ∇(u − uN ) L2 (Ω) ≤ Cp2 (1 + ln p) e−bλp + εp3 e−b L . Furthermore, ip,Γ uε = uN |Γ ∀ edges Γ of T (min{1, λpε}, L) with Γ ⊂ ∂Ω. In particular, if the number of elements |T (min {λpε, 1}, L)| ∼ L ∼ p, then N = dim S p (T (min {λpε, 1}, L)) ∼ p3 and therefore u − uN ε ≤ Ce−b N 1/3 , where the constants C, b are independent of ε (albeit dependent on λ). 8. 8 is involved.
J − 1} either Ki ⊂ Bκ (Aj ) ∩ Ωj or Ki ⊂ Bκ (Aj ) ∩ Ωj+1 . Additionally, denoting hi = diam Ki , the element map Mi satisfies Dα s˜−1 j,hi ◦ Mi (˜ s−1 j,hi ◦ Mi ) ˆ i) L∞ (K ≤ Cγ |α| α! ˆ i) L∞ (K ≤ C; −1 ∀α ∈ N20 , furthermore, exactly one of the following situations is satisfied: either Aj ∈ Ki and hi ≤ c4 κσ L or Aj ∈ Ki and c3 hi ≤ dist(Aj , Ki ) ≤ c4 hi . , Ki ⊂ Ω \ Uc2 κ (∂Ω), and the element map satisfies D α Mi −1 (Mi ) ˆ i) L∞ (K ≤ Cγ |α| α! ˆ i) L∞ (K ≤ ∀α ∈ N20 , C . 8, we will choose κ = O(pε), where p is the approximation order.