By Thomas Apel, Olaf Steinbach
This quantity on a few fresh elements of finite aspect tools and their functions is devoted to Ulrich Langer and Arnd Meyer at the celebration 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 large variety in their pursuits within the conception and alertness of finite point methods.
Topics conceal the research of area decomposition and multilevel tools, together with hp finite parts, hybrid discontinuous Galerkin tools, and the coupling of finite and boundary point tools; the effective answer of eigenvalue difficulties concerning 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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Define the face-to-element extension EF→T and the element-to-face restriction operator RT →F between the reference tetrahedron T and the reference face F from (2), (3) as (EF→T u) (x, y, z) = u(x, y + z), (19) 1 (RT →F w) (x, y) = w(x, sy, (1 − s)z) ds. (20) 0 These operators are mappings between P p (T ) and P p (F), preserve the function on the edge y = z = 0, and are continuous with respect to the norms RT →F w EF→T u L2 (F),y L2 (T ) w u L2 (T ) , RT →F w L2 (F),y , EF→T u w H 1 (F),y H 1 (T ) u H 1 (T ) , H 1 (F),y with the norms u 2 L2 (F),y yu(x, y)2 d(x, y), := F u 2 H 1 (F),y := u 2 L2 (F),y + ∇u 2 L2 (F),y .
But little else could have been expected from 8 MB memory per processor which is less than the L3-cache on recent server processors. 1 State of Hardware Let us jump to the parallel computers that are locally available to the former members of the numerical mathematics group. We restrict ourselves to the locations Chemnitz, Linz and Graz. 21 Tflops sustained performance. A 10 Gbit InfiniBand is used as interconnection network. The cluster was delivered from Megware, Chemnitz. The CHiC project is conducted by a consortium of scientists from multiple departments of the university: the departments of Computer Science, Mathematics, Mechanical Engineering, Natural Sciences, Electrical Engineering and Information Technology, Humanities, and other institutes.
We think of operator splitting methods, where one has to solve a large scale symmetric matrix equation in each time-step. In recent years hybridization methods appeared, which allow to reduce the discrete system to the element interfaces . This paper is concerned with the construction and analysis of domain decomposition methods for the Hybrid Discontinuous Galerkin (HDG) method. We consider one element as sub-domain, and the coarse grid problem consists of mean values on element interfaces.