By Jens M. Melenk

Many partial differential equations bobbing up in perform are parameter-dependent difficulties which are of singularly perturbed variety. well known examples contain plate and shell types for small thickness in reliable mechanics, convection-diffusion difficulties in fluid mechanics, and equations coming up in semi-conductor gadget modelling. universal beneficial properties of those difficulties are layers and, in relation to non-smooth geometries, nook singularities. Mesh layout ideas for the effective approximation of either beneficial properties by way of the hp-version of the finite point technique (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 line with those mesh layout rules is demonstrated conscientiously.

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The results of this work can be extended to the case of piecewise analytic functions A, c, f . 6. 14 holds with the weight function Φp,β,ε replaced with Φp,β,ε ≡ 1, [95]. 7. 11) is a Dirichlet problem. 4 for Neumann, Robin boundary conditions, and a transmission problem. 1 does not capture the boundary layer character of the solution. This is done classically with the aid of asymptotic expansions. Since additionally corner singularities are present in the solution uε , we present next a decomposition of the solution uε into a smooth part wε , a CL boundary layer part uBL ε , a corner layer part uε , and a small remainder rε .

2 shows that this situation is met in practice. Although the high accuracy of hp-ﬁnite element methods is their most striking feature, there are additional reasons for their increasing popularity, especially in solid mechanics. We mention here the issue of numerical locking in parameterdependent problems, where high order methods tend to be more robust than their low-order counterparts, [24]. The hardest problems in linear ﬁnite element analysis where locking problems are rampant are shell problems.

First we introduce the weight functions Φp,β,ε as follows. With each vertex Aj , j = 1, . . , J, we associate a number βj ∈ [0, 1), set β = (β1 , . . , βJ ), and write for p ∈ N0 ˆp,β ,ε (x) := Φ j min 1, dist(x, Aj ) min {1, ε(p + 1)} p+βj . 3 Regularity: the two-dimensional case 33 The weight function Φp,β,ε is then deﬁned as J ˆp,β ,ε (x). 1 (Regularity in countably normed spaces). 4). , uε Φp,β,ε ∇ p+2 uε ε L2 (Ω) ≤C ≤ CK p max {p + 1, ε−1 }p+2 ∀p ∈ N0 . In particular, for ﬁxed neighborhoods Uj of the vertices Aj there holds for all p ∈ N0 and all x ∈ Uj ∩ Ω |∇p (u(x) − u(Aj )) | ≤ CK p ε−1 min 1, rj ε 1−βj rj−p max p + 1, rj ε p+1 , where rj := dist(x, Aj ).