By Thomas Apel, Olaf Steinbach

This quantity on a few fresh elements of finite point equipment and their functions is devoted to Ulrich Langer and Arnd Meyer at the party in their sixtieth birthdays in 2012. Their paintings combines the numerical research of finite aspect algorithms, their effective implementation on state-of-the-art architectures, and the collaboration with engineers and practitioners. during this spirit, this quantity includes contributions of former scholars and collaborators indicating the vast diversity in their pursuits within the idea and alertness of finite point methods.

Topics hide the research of area decomposition and multilevel equipment, together with hp finite parts, hybrid discontinuous Galerkin equipment, and the coupling of finite and boundary aspect equipment; the effective answer of eigenvalue difficulties with regards to partial differential equations with purposes in electric engineering and optics; and the answer of direct and inverse box difficulties in reliable mechanics.

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**Example text**

6 Historic Numerical Examples Let us present two examples from the early 1990s published in [26] where parallel hierarchical preconditioners have been applied. Detailed numerical investigations of many additive and multiplicative Schwarz preconditioners can be found in [26]. We considered the Poisson equation in a domain Ω depicted in Fig. 9 with mixed Dirichlet and Neumann boundary conditions. The appropriate results on the MultiCluster-2 can be found in Tab. 3. x2 ✻ Ω1 ΓD Ω Ω t 2t t t t 3t t t ˆ ”I” ❅ ❅❅ ❅❅ ❅❅ t ❅ ❅❅ t ❅ ❅❅ t ❅ ❅ t {·} = ❅❅❅ ❅❅❅ ❅❅❅ ❅ t❅ ❅❅❅❅ t❅ ❅❅❅❅ t❅ ❅❅❅ t ❅ t❅ ❅ ❅❅❅ t❅ ❅ ❅❅❅ t❅ ❅ ❅❅ t {•} = ˆ ”C” ❅ ❅ ❅ t ❅ t ❅ t ❅ t ❅ t ❅ t ❅ t ❅ t ❅ t ❅ t ❅ t ❅ t ❅ t 2 ❅ ❅❅ ❅ ❅❅ t ❅ ❅t t ❅ ❅t ❅❅❅ ❅❅❅ 8 ❅ ❅ t❅ ❅❅❅ tΩ5 Ω = Ω i Ω4 t❅ ❅❅❅t ❅ ❅ i=1 t❅ ❅ ❅❅t t❅ ❅ ❅❅ t ❅ ❅ t t t t t t t t t t ❅ ❅ ❅ ❅ t t ❅ ❅ ❅ ❅ t ΓD = {(x, 0) : 0 ≤ x ≤ 3} 1 ❅ ❅ ❅ ❅ ❅ ❅ ❅ t ❅ ❅❅ t ❅ ❅❅ t ❅ ❅t ∪ {(x, 3) : 0 ≤ x ≤ 1} ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ t❅ ❅❅❅❅ t❅ ❅❅❅❅ t❅ ❅❅❅ t 8 ❅ t❅ ❅ ❅❅❅ t❅ ❅ ❅❅❅ t❅ ❅ ❅❅ t ΓC = ∂ Ω i \ ΓD ❅ i=1 ❅ ❅ ❅ ❅❅ ❅ ❅ ❅ ❅❅ ❅❅❅❅ ✲ 0 3 0 Ω6 1 Ω7 2 Ω8 ΓD x1 3 Fig.

Since wi p ∑k=i wk 1 , p−i+1 we get 1 (1 − y)2i−1dip (y)2 dy 0 1 1 (p − i + 1)2 1 p (1 − y)2i−1ei+1 (y)2 dy 2 (1 − y)2i−1l 2i−1 p−i (y) dy + 0 0 1 . (p − i + 1)3 p The additional factor similarly. i2 p2 follows trivially since p/2 ≤ i ≤ p. Estimate (18) follows 5 Extension from a Vertex In this section we define and analyze minimal extensions from a vertex of the reference tetrahedron T . Lemma 5. We define 1 e˜V = y2 v(y)2 dy argmin v∈P p−1 ,v(0)=1 0 and eV (y) = (1 − y)e˜V (y). 44 J. Sch¨oberl and C.

Sch¨oberl and C. Lehrenfeld Proof. Let λ be double-valued on faces with consistent mean-values, this means λ = (λT )T ∈T ∈ ∏ P p (FT ), T ∈T such that λT = λT F for F = T ∩ T . F Define the average λ˜ ∈ P p (F ) as ∑T :F⊂T λT |F λ˜ = . e. λ˜ where λT 2 S,T λ˜ 2 S F = ≤ ≤ c(p) ∇u := infu∈P p (T ) We use F λ˜ = 2 S ∑ λT 2 S,T , u−λ 2 j,∂ T T ∈T 2 L2 (T ) + . λT to apply Theorem 3 for estimating ∑ T ∈T ∑ T ∈T λ˜ |∂ T ∑ 2 S,T T ∈T λT 2S,T (log p)γ (log p)γ (log p)γ + ∑ F∈FT ∑ λT ∑ λT T ∈T T ∈T ∑ T ∈T λT λT 2 S,T + λ˜ |∂ T − λT λ˜ ∂ T − λT 2 S,T + 2 S,T + 2 F,0 ∑ λ˜ ∂ T − λT ∑ λT F∈FT F∈FT 2 S,T 2 F 2 F 2 S,T .