SUMMARYIn this paper we present a new approach to the parallelization of algebraic multigrid (AMG), i.e., to the parallel coarse grid selection in AMG. Our approach does not involve any special treatment of processor subdomain boundaries and hence avoids a number of drawbacks of other AMG parallelization techniques. The key idea is to select an appropriate (local) coarse grid on each processor from all admissible grids such that the composed coarse grid forms a suitable coarse grid for the whole domain, i.e. there is no need for any boundary treatment. To this end, we first construct multiple equivalent coarse grids on each processor subdomain. In a second step we then select exactly one grid per processor by a graph clustering technique. The results of our numerical experiments clearly indicate that this approach results in coarse grids of high quality which are very close to those obtained with sequential AMG. Furthermore, the operator and grid complexities of our parallel AMG are mostly smaller than those obtained by other parallel AMG methods, whereas the scale-up behavior of the proposed algorithm is similar to that of other parallel AMG techniques. However a significant improvement with respect to the speed-up performance is achieved.

The consideration of geomechanical effects is becoming more and more important in reservoir simulations. Ensuring stable simulation processes often enough requires handling the entire process with all types of physical unknowns fully implicitly. However, the resulting fully coupled linear systems pose challenges for linear solvers. The number of approaches that can efficiently handle a fully coupled system is extremely limited. System-AMG has demonstrated its efficiency for isothermal and thermal reservoir simulations. At the same time, AMG is known to be a robust and highly efficient linear solver for mere linear elasticity problems. This paper will discuss the combination of the advantages that AMG approaches have for both types of physics. This results in a robust and efficient solution scheme for the fully coupled linear system. The Automatic Differentiation General Purpose Research Simulator (AD-GPRS) is used to produce the Jacobians that are guaranteed to be exact. In a single-phase case, the overall Jacobian matrix takes the form of a constrained linear elasticity system where the flow unknowns serve as a Lagrangian multiplier. In other words, a saddle point system needs to be solved, where the flow and the mechanics problem might come at very different scales. A natural relaxation method for this kind of systems is given by Uzawa smoothing schemes which provide a way to overcome the difficulties that other smoothers may encounter. This approach appears intuitive for single-phase problems, where Gauss-Seidel can be applied in an inexact Uzawa scheme. However, in the multiphase case, incomplete factorization smoothers are required for the flow and transport part. We will discuss the incorporation in an inexact Uzawa scheme, where different realizations are possible, with different advantages and disadvantages. Finally, we propose an adaptive mechanism along with the outer Krylov solver to detect the best-suited realization for a given linear system. In the multiphase case, also the matrix preprocessing, for instance, by Dynamic Row Summing, needs to be considered. However, the process now also needs to reflect the requirements of the Uzawa scheme to be applicable. We demonstrate the performance for widely used test cases as well as for real-world problems of practical interest.

Baryonic excitation spectra, electroweak and strong decay properties are discussed within a relativistically covariant constituent quark model based on the instantaneous approximation to the three-body Bethe-Salpeter equation.

We investigate algebraic multigrid (AMG) methods for the linear systems arising from the discretization of Navier-Stokes equations via the finite pointset method. In the segregated approach, three pressure systems and one velocity system need to be solved. In the coupled approach, one of the pressure systems is coupled with the velocity system, leading to a coupled velocity-pressure saddle point system. The discretization of the differential operators used in FPM leads to non-symmetric matrices that do not have the M-matrix property. Even though the theoretical framework for many AMG methods requires these properties, our AMG methods can be successfully applied to these matrices and show a robust and scalable convergence when compared to a BiCGStab(2) solver. Keywords Algebraic multigrid • Finite pointset method • Meshfree method • Saddle point problem Communicated by Gabriel Wittum.

We investigate various block preconditioners for a low-order Raviart-Thomas discretization of the mixed Poisson problem on adaptive quadrilateral meshes. In addition to standard diagonal and Schur complement preconditioners, we present a dedicated AMG solver for saddle point problems (SPAMG). A key element is a stabilized prolongation operator that couples the flux and scalar components. Our numerical experiments in 2D and 3D show that the SPAMG preconditioner displays nearly mesh-independent iteration counts for adaptive meshes and heterogeneous coefficients.