Complex additive geometric multilevel solvers for Helmholtz equations on spacetrees

We introduce a family of implementations of low-order, additive, geometric multilevel solvers for systems of Helmholtz equations arising from Schrödinger equations. Both grid spacing and arithmetics may comprise complex numbers, and we thus can apply complex scaling to the indefinite Helmholtz operator. Our implementations are based on the notion of a spacetree and work exclusively with a finite number of precomputed local element matrices. They are globally matrix-free. Combining various relaxation factors with two grid transfer operators allows us to switch from additive multigrid over a hierarchical basis method into a Bramble-Pasciak-Xu (BPX)-type solver, with several multiscale smoothing variants within one code base. Pipelining allows us to realize full approximation storage (FAS) within the additive environment where, amortized, each grid vertex carrying degrees of freedom is read/written only once per iteration. The codes realize a single-touch policy. Among the features facilitated by matrix-free FAS is arbitrary dynamic mesh refinement (AMR) for all solver variants. AMR as an enabler for full multigrid (FMG) cycling—the grid unfolds throughout the computation—allows us to reduce the cost per unknown. The present work primary contributes toward software realization and design questions. Our experiments show that the consolidation of single-touch FAS, dynamic AMR, and vectorization-friendly, complex scaled, matrix-free FMG cycles delivers a mature implementation blueprint for solvers of Helmholtz equations in general. For this blueprint, we put particular emphasis on a strict implementation formalism as well as some implementation correctness proofs

Seventh Copper Mountain Conference on Multigrid Methods

The Seventh Copper Mountain Conference on Multigrid Methods was held on April 2-7, 1995 at Copper Mountain, Colorado. This book is a collection of many of the papers presented at the conference and so represents the conference proceedings. NASA Langley graciously provided printing of this document so that all of the papers could be presented in a single forum. Each paper was reviewed by a member of the conference organizing committee under the coordination of the editors. The vibrancy and diversity in this field are amply expressed in these important papers, and the collection clearly shows the continuing rapid growth of the use of multigrid acceleration techniques

The Sixth Copper Mountain Conference on Multigrid Methods, part 2

The Sixth Copper Mountain Conference on Multigrid Methods was held on April 4-9, 1993, at Copper Mountain, Colorado. This book is a collection of many of the papers presented at the conference and so represents the conference proceedings. NASA Langley graciously provided printing of this document so that all of the papers could be presented in a single forum. Each paper was reviewed by a member of the conference organizing committee under the coordination of the editors. The multigrid discipline continues to expand and mature, as is evident from these proceedings. The vibrancy in this field is amply expressed in these important papers, and the collection clearly shows its rapid trend to further diversity and depth

Schnelle Löser für Partielle Differentialgleichungen

This workshop was well attended by 52 participants with broad geographic representation from 11 countries and 3 continents. It was a nice blend of researchers with various backgrounds

Adaptive multilevel solvers for the discontinuous Petrov–Galerkin method with an emphasis on high-frequency wave propagation problems

This dissertation focuses on the development of fast and efficient solution schemes for the simulation of challenging problems in wave propagation phenomena. In particular, emphasis is given on high frequency acoustic and electromagnetic problems which are characterized by localized solutions. This kind of simulations are essential in various applications, such as ultrasonic testing, laser scanning and modeling of optical laser amplifiers. In wave simulations, the computational cost of any numerical method, is directly related to the frequency. In the high-frequency regime very fine meshes have to be used in order to satisfy the Nyquist criterion and overcome the pollution effect. This often leads to prohibitively expensive problems. Numerical methods based on standard Galerkin discretizations lack pre-asymptotic discrete stability and therefore adaptive mesh refinement strategies are usually inefficient. Additionally, the indefinite nature of the wave operator makes state of the art preconditioning techniques, such as multigrid, unreliable. In this work, a promising alternative approach is followed within the framework of the discontinuous Petrov–Galerkin (DPG) method. The DPG method offers numerous advantages for our problems of interest. First and foremost, it offers mesh and frequency independent discrete stability even in the pre-asymptotic region. This is made possible by computing, on the fly, an optimal test space as a function of the trial space. Secondly, it provides a built-in local error indicator that can be used to drive adaptive refinements. Combining these two properties together, reliable adaptive refinement strategies are possible which can be initiated from very coarse meshes. Lastly, the DPG method can be viewed as a minimum residual method, and therefore it always delivers symmetric (Hermitian) positive definite stiffness matrix. This is a desirable advantage when it comes to the design of iterative solution algorithms. Conjugate Gradient based solvers can be employed which can be accelerated by domain decomposition (one- or multi- level) preconditioners for symmetric positive definite systems. Driven by the aforementioned properties of the DPG method, an adaptive multigrid preconditioning technology is developed that is applicable for a wide range of boundary value problems. Unlike standard multigrid techniques, our preconditioner involves trace spaces defined on the mesh skeleton, and it is suitable for adaptive hp-meshes. Integration of the iterative solver within the DPG adaptive procedure turns out to be crucial in the simulation of high frequency wave problems. A collection of numerical experiments for the solution of linear acoustics and Maxwell equations demonstrate the efficiency of this technology, where under certain circumstances uniform convergence with respect to the mesh size, the polynomial order and the frequency can be achieved. The construction is complemented with theoretical estimates for the condition number in the one-level setting.Computational Science, Engineering, and Mathematic

KSPHPDDM and PCHPDDM: Extending PETSc with advanced Krylov methods and robust multilevel overlapping Schwarz preconditioners

[EN] Contemporary applications in computational science and engineering often require the solution of linear systems which may be of different sizes, shapes, and structures. The goal of this paper is to explain how two libraries, PETSc and HPDDM, have been interfaced in order to offer end-users robust overlapping Schwarz preconditioners and advanced Krylov methods featuring recycling and the ability to deal with multiple right-hand sides. The flexibility of the implementation is showcased and explained with minimalist, easy-to-run, and reproducible examples, to ease the integration of these algorithms into more advanced frameworks. The examples provided cover applications from eigenanalysis, elasticity, combustion, and electromagnetism.Jose E. Roman was supported by the Spanish Agencia Estatal de Investigacion (AEI) under project SLEPc-DA (PID2019-107379RB-I00)Jolivet, P.; Roman, JE.; Zampini, S. (2021). KSPHPDDM and PCHPDDM: Extending PETSc with advanced Krylov methods and robust multilevel overlapping Schwarz preconditioners. Computers & Mathematics with Applications. 84:277-295. https://doi.org/10.1016/j.camwa.2021.01.0032772958