By Thomas Apel, Olaf Steinbach

This quantity on a few contemporary points of finite point equipment and their functions is devoted to Ulrich Langer and Arnd Meyer at the get together in their sixtieth birthdays in 2012. Their paintings combines the numerical research of finite aspect algorithms, their effective implementation on cutting-edge architectures, and the collaboration with engineers and practitioners. during this spirit, this quantity comprises contributions of former scholars and collaborators indicating the vast variety in their pursuits within the idea and alertness of finite aspect methods.

Topics disguise the research of area decomposition and multilevel equipment, together with hp finite components, hybrid discontinuous Galerkin equipment, and the coupling of finite and boundary aspect tools; the effective answer of eigenvalue difficulties concerning partial differential equations with functions in electric engineering and optics; and the answer of direct and inverse box difficulties in sturdy mechanics.

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Hierarchical extension operators and local multigrid methods in domain decomposition preconditioners. East-West J. Numer. Math. : Bibliotheken zur Entwicklung paralleler Algorithmen. : Adaptive domain decomposition methods for finite and boundary element equations. L. ) Boundary Element Topics. Reports from the Final Conference of the Priority Research Programme of the German Research Foundation (DFG), Stuttgart, October 2-4, 1995, pp. 121–148. : A Parallel Algebraic Multigrid Solver on Graphics Processing Units.

Parallel Algorithms for Partial Differential Equations. Proceedings of the Sixth GAMM-Seminar, Kiel, January 19-21,1990. : On the fast matrix multiplication in the boundary element method by panel clustering. Numer. Math. : Multigrid-Newton-methods for the calculation of electromagnetic fields. In: [50], pp. : Nichtlineare Berechnung station¨arer Magnetfelder einer Gleichstrommaschine mittels Full-Multigrid-Newton-Techniken. In: [51], pp. 135–146 (1990) [35] Hengst, S. ): Proceedings of the ”5-th Multigrid Seminar” held at Eberswalde, May 14-18.

Proof. We convert the extended function into the Legendre basis as p EE→F u = ∑ Pi i=0 x (1 − y)i vi (y), 1−y where vi ∈ P p−i are vi (y) = p p ui ei (y) ui+2 (1 − y)2ei+2 (y) − . 2i − 1 2i + 3 We rewrite ui ui+2 ui+2 p − e p (y) + eip (y) − (1 − y)2ei+2 (y) 2i − 1 2i + 3 i 2i + 3 ui+2 p = vi (0)eip (y) + d p (y) + (1 − y)di+1 (y) . 2i + 3 i vi (y) = (25) p Note that there holds u(x) = ∑i=0 vi (0)Pi (x). From Lemma 4 and Lemma 7 there follows EE→F u 2 L2 (F),y 1 p 1 ∑ i+1 i=0 p y(1 − y)2i+1 eip (y)vi (0) + dip (y) + (1 − y)di+1 (y) 0 ui+2 2i + 3 2 dy 50 J.