In this article an unfitted Discontinuous Galerkin Method is proposed to discretize elliptic interface problems. The method is based on the Symmetric Interior Penalty Discontinuous Galerkin Method and can also be interpreted as a generalization of the method given in [A. Hansbo, P. Hansbo, An unfitted finite element method based on Nitsche's method for elliptic interface prob
Adaptive multiscale methods are among the many effective techniques for the numerical solution of partial differential equations. Efficient grid management is an important task in these solvers. In this paper we focus on this problem for discontinuous Galerkin discretization methods in 2 and 3 spatial dimensions and present a data structure for handling adaptive grids of different cell types in a unified approach. Instead of tree-based techniques where connectivity is stored via pointers, we associate each cell that arises in the refinement hierarchy with a cell identifier and construct algorithms that establish hierarchical and spatial connectivity. By means of bitwise operations, the complexity of the connectivity algorithms can be bounded independent of the level. The grid is represented by a hash table which results in a low-memory data structure and ensures fast access to cell data. The spatial connectivity algorithm also supports the application of quadrature rules for face integrals that occur in discontinuous Galerkin discretizations. The concept allows us to implement discontinuous Galerkin methods largely independent of spatial dimension and cell type. We demonstrate this by outlining how typical algorithmic tasks that arise in these implementations can be performed with our data structure. In computational tests we compare our approach with that of a classical implementation using pointers.
The new flow solver Quadflow, developed within the SFB 401, has been designed for investigating flows around airfoils and simulating the interaction of the structural dynamics and aerodynamics. This article addresses the following issues arising in this context. After identifying proper coupling conditions and settling the well-posedness of the resulting coupled fluid-structure problem, suitable strategies for successively applying flow and structure solvers needed to be developed that give rise to a sufficiently close coupling of both media. Based on these findings the overall efficiency of numerical simulations hinges, for the current choice of structural models, on the efficiency of the flow solver. In addition to the multiscale-based grid adaptation concepts, proper parallelization concepts are needed to realize for such complex problems an acceptable computational performance on parallel architectures. Since the parallelization of dynamically varying adaptive discretizations is by far not straightforward we will mainly concentrate on this issue in connection with the above mentioned multiscale adaptivity concepts. In particular, we outline the way the multiscale library has been parallelized via MPI for distributed memory architectures. To ensure a proper scaling of the computational performance with respect to CPU time and memory, the load of data has to be well-balanced and communication between processors has to be minimized. We point out how to meet these requirements by employing the concept of space-filling curves.
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