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1. Optimal Geometric Multigrid Preconditioners for HDG-P0 Schemes for the reaction-diffusion equation and the Generalized Stokes equations

   https://doi.org/10.1051/m2an/2023025  

   Fu, Guosheng; Kuang, Wenzheng (May 2023, ESAIM: Mathematical Modelling and Numerical Analysis)

   We present the lowest-order hybridizable discontinuous Galerkin schemes with numerical integration (quadrature), denoted as HDG-P0 for the reaction-diffusion equation and the generalized Stokes equations on conforming simplicial meshes in two- and three-dimensions. Here by lowest order, we mean that the (hybrid) finite element space for the global HDG facet degrees of freedom (DOFs) is the space of piecewise constants on the mesh skeleton. A discontinuous piecewise linear space is used for the approximation of the local primal unknowns. We give the optimal a priori error analysis of the proposed HDG-P0 schemes, which hasn’t appeared in the literature yet for HDG discretizations as far as numerical integration is concerned. Moreover, we propose optimal geometric multigrid preconditioners for the statically condensed HDG-P0 linear systems on conforming simplicial meshes. In both cases, we first establish the equivalence of the statically condensed HDG system with a (slightly modified) nonconforming Crouzeix–Raviart (CR) discretization, where the global (piecewise-constant) HDG finite element space on the mesh skeleton has a natural one-to-one correspondence to the nonconforming CR (piecewise-linear) finite element space that live on the whole mesh. This equivalence then allows us to use the well-established nonconforming geometry multigrid theory to precondition the condensed HDG system. Numerical results in two- and three-dimensions are presented to verify our theoretical findings. 

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2. Dispersion analysis of finite difference and discontinuous Galerkin schemes for Maxwell’s equations in linear Lorentz media

   Jiang, Yan; Sakkaplangkul, Puttha; Bokil, Vrushali A; Cheng, Yingda; Li, Fengyan (May 2019, Journal of computational physics)

   In this paper, we consider Maxwell’s equations in linear dispersive media described by a single-pole Lorentz model for electronic polarization. We study two classes of commonly used spatial discretizations: finite difference methods (FD) with arbitrary even order accuracy in space and high spatial order discontinuous Galerkin (DG) finite element methods. Both types of spatial discretizations are coupled with second order semi-implicit leap-frog and implicit trapezoidal temporal schemes. By performing detailed dispersion analysis for the semi-discrete and fully discrete schemes, we obtain rigorous quantification of the dispersion error for Lorentz dispersive dielectrics. In particular, comparisons of dispersion error can be made taking into account the model parameters, and mesh sizes in the design of the two types of schemes. This work is a continuation of our previous research on energy-stable numerical schemes for nonlinear dispersive optical media [6,7]. The results for the numerical dispersion analysis of the reduced linear model, considered in the present paper, can guide us in the optimal choice of discretization parameters for the more complicated and nonlinear models. The numerical dispersion analysis of the fully discrete FD and DG schemes, for the dispersive Maxwell model considered in this paper, clearly indicate the dependence of the numerical dispersion errors on spatial and temporal discretizations, their order of accuracy, mesh discretization parameters and model parameters. The results obtained here cannot be arrived at by considering discretizations of Maxwell’s equations in free space. In particular, our results contrast the advantages and disadvantages of using high order FD or DG schemes and leap-frog or trapezoidal time integrators over different frequency ranges using a variety of measures 

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3. Multigrid for model reduction of power grid networks

   https://doi.org/10.1002/nla.2201  

   Lee, B. (June 2018, Numerical Linear Algebra with Applications)

   Summary This paper presents a method for determining the relevant buses for reduced models of power grid networks described by systems of differential‐algebraic equations and for constructing the coarse‐grain dynamical power grid systems. To determine these buses, time integration of differential equations is not needed, but rather, a stationary system is analyzed. However, unlike stationary‐system approaches that determine only coarse generator buses by approximating the coherency of the generators, the proposed method analyzes the graph Laplacian associated with the admittance matrix. The buses for the reduced model are chosen to ensure that the graph Laplacian of the reduced model is an accurate approximation to the graph Laplacian of the full system. Both load and generator buses can be selected by this procedure since the Laplacian is defined on all the buses. The basis of this proposed approach lies in the close relationship between the synchrony of the system and the spectral properties of this Laplacian, that is, conditions on the spectrum of this Laplacian that almost surely guarantee the synchrony of the system. Thus, assuming that the full system is in synchrony, our strategy is to coarsen the full‐system Laplacian such that the coarse Laplacian possesses good approximation to these spectral conditions. Accurate approximation to these conditions then can better lead to synchronous reduced models. The coarsened Laplacian is defined on coarse degrees of freedom (DOFs), which are associated with the relevant buses to include in the reduced model. To realize this coarse DOF selection, we use multigrid coarsening techniques based on compatible relaxation. Multigrid is the natural choice since it has been extensively used to coarsen Laplacians arising from discretizations of elliptic partial differential equations and is actively being extended to graph Laplacians. With the selection of the buses for the reduced model, the reduced model is completed by constructing the coarse admittance matrix values and other physical parameters using standard power grid techniques or by using the intergrid operators constructed in the coarse DOFs selection process. Unfortunately, the selection of the coarse buses and the coarsening of the admittance matrix and physical parameters are not sufficient by themselves to produce a stable reduced system. To achieve a stable system, system structures of the fine‐grain model must be preserved in the reduced model. We analyze this to develop a multigrid methodology for constructing stable reduced models of power grid systems. Numerical examples are presented to validate this methodology. 

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4. Algebraic multigrid for the nonlinear powerflow equations

   https://doi.org/10.1002/nla.2347  

   Lee, Barry; Pereira Batista, Enrique (November 2020, Numerical Linear Algebra with Applications)

   Abstract In a recent article, one of the authors developed a multigrid technique for coarse‐graining dynamic powergrid models. A key component in this technique is a relaxation‐based coarsening of the graph Laplacian given by the powergrid network and its weighted graph, which is represented by the admittance matrix. In this article, we use this coarsening strategy to develop a multigrid method for solving a static system of nonlinear equations that arises through Ohm's law, the so‐called powerflow equations. These static equations are tightly knitted to the dynamic model in that the full powergrid model is an algebraic‐differential system with the powerflow equations describing the algebraic constraints. We assume that the dynamic model corresponds to a stable operating powergrid, and thus, the powerflow equations are associated with a physically stable system. This stability permits the coarsening of the powerflow equations to be based on an approximate graph Laplacian, which is embedded in the powerflow system. By algebraically constructing a hierarchy of approximate weighted graph Laplacians, a hierarchy of nonlinear powerflow equations immediately becomes apparent. This latter hierarchy can then be used in a full approximation scheme (FAS) framework that leads to a nonlinear solver with generally a larger basin of attraction than Newton's method. Given the algebraic multigrid (AMG) coarsening of the approximate Laplacians, the solver is an AMG‐FAS scheme. Alternatively, using the coarse‐grid nodes and interpolation operators generated for the hierarchy of approximate graph Laplacians, a multiplicative‐correction scheme can be derived. The derivation of both schemes will be presented and analyzed, and numerical examples to demonstrate the performance of these schemes will be given. 

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5. Scalable DPG multigrid solver with applications in high-frequency wave propagation

   https://doi.org/10.26153/tsw/55685  

   Badger, Jacob (January 2024, The University of Texas at Austin)

   Wave propagation is fundamental to applications including natural resource exploration, nuclear fusion research, and military defense, among others. However, developing accurate and efficient numerical algorithms for solving time-harmonic wave propagation problems is notoriously difficult. One difficulty is that classical discretization techniques (e.g., Galerkin finite elements, finite difference, etc.) yield indefinite discrete systems that preclude the use of many scalable solution algorithms. Significant progress has been made to develop specialized preconditioners for high-frequency wave propagation problems but robust and scalable solvers for general problems, including non-homogenous media and complex geometries, remain elusive. An alternative approach is to use minimum residual discretization methods—that yield Hermitian positive-definite discrete systems—and may be amenable to more standard preconditioners. Indeed, popularization of the first-order system least-squares methodology (FOSLS) was driven by the applicability of geometric and algebraic multigrid to otherwise indefinite problems. However, for wave propagation problems, FOSLS is known to be highly dissipative and is thus less competitive in the high-frequency regime. The discontinuous Petrov–Galerkin (DPG) method of Demkowicz and Gopalakrishnan is a minimum residual finite element method with several additional attractive properties: mesh-independent stability, a built-in error indicator, and applicability to a number of variational formulations. In the context of high-frequency wave propagation, the ultraweak DPG formulation has been observed to produce pollution error roughly commensurate to Galerkin discretizations. DPG discretizations may thus deliver accuracy typical of classical discretization techniques, but result in Hermitian positive-definite discrete systems that are often more amenable to preconditioning. A multigrid preconditioner for DPG systems, developed in the dissertation work of S. Petrides, was shown to scale efficiently in a shared-memory implementation. The primary objective of this dissertation is development of an efficient, distributed implementation of the DPG multigrid solver (DPG-MG). The distributed DPG-MG solver developed in this work will be demonstrated to be massively scalable, enabling solution of three-dimensional problems with O(10¹²) degrees of freedom on up to 460 000 CPU cores, an unprecedented scale for high-frequency wave propagation. The scalability of the DPG-MG solver will be further combined with hp-adaptivity to enable efficient solution of challenging real-world high-frequency wave propagation problems including optical fiber modeling, simulation of RF heating in tokamak devices, and seismic simulation. These applications include complex three-dimensional geometries, heterogeneous and anisotropic media, and localized features; demonstrating the robustness and versatility of the solver and tools developed in this dissertation. 

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