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This work describes the development of matrix-free GPU-accelerated solvers for high-order finite element problems in if(div). The solvers are applicable to grad-div and Darcy problems in saddle-point formulation, and have applications in radiation diffusion and porous media flow problems, among others. Using the interpolation--histopolation basis (cf. [W. Pazner, T. Kolev, and C. R. Dohrmann, SIAM J. Sci. Comput., 45 (2023), pp. A675-A702]), efficient matrix-free preconditioners can be constructed for the (1,1)-block and Schur complement of the block system. With these approximations, block-preconditioned MINRES converges in a number of iterations that is independent of the mesh size and polynomial degree. The approximate Schur complement takes the form of an M-matrix graph Laplacian and therefore can be well-preconditioned by highly scalable algebraic multigrid methods. High-performance GPU-accelerated algorithms for all components of the solution algorithm are developed, discussed, and benchmarked. Numerical results are presented on a number of challenging test cases, including the "crooked pipe" grad-div problem, the SPE10 reservoir modeling benchmark problem, and a nonlinear radiation diffusion test case. [ABSTRACT FROM AUTHOR]

In this article, we present algorithms and implementations for the end-to-end GPU acceleration of matrix-free low-order-refined preconditioning of high-order finite element problems. The methods described here allow for the construction of effective preconditioners for high-order problems with optimal memory usage and computational complexity. The preconditioners are based on the construction of a spectrally equivalent low-order discretization on a refined mesh, which is then amenable to, for example, algebraic multigrid preconditioning. The constants of equivalence are independent of mesh size and polynomial degree. For vector finite element problems in H (curl) and H (div) (e.g., for electromagnetic or radiation diffusion problems), a specially constructed interpolation–histopolation basis is used to ensure fast convergence. Detailed performance studies are carried out to analyze the efficiency of the GPU algorithms. The kernel throughput of each of the main algorithmic components is measured, and the strong and weak parallel scalability of the methods is demonstrated. The different relative weighting and significance of the algorithmic components on GPUs and CPUs is discussed. Results on problems involving adaptively refined nonconforming meshes are shown, and the use of the preconditioners on a large-scale magnetic diffusion problem using all spaces of the finite element de Rham complex is illustrated. [ABSTRACT FROM AUTHOR]

In this paper we present a unified framework for constructing spectrally equivalent low-order-refined discretizations for the high-order finite element de Rham complex. This theory covers diffusion problems in H1, \bfitH (curl), and \bfitH (div) and is based on combining a low-order discretization posed on a refined mesh with a high-order basis for N\'ed\'elec and Raviart--Thomas elements that makes use of the concept of polynomial histopolation (polynomial fitting using prescribed mean values over certain regions). This spectral equivalence, coupled with algebraic multigrid methods constructed using the low-order discretization, results in highly scalable matrix-free preconditioners for high-order finite element problems in the full de Rham complex. Additionally, a new lowestorder (piecewise constant) preconditioner is developed for high-order interior penalty discontinuous Galerkin (DG) discretizations, for which spectral equivalence results and convergence proofs for algebraic multigrid methods are provided. In all cases, the spectral equivalence results are independent of polynomial degree and mesh size; for DG methods, they are also independent of the penalty parameter. These new solvers are flexible and easy to use; any "black-box" preconditioner for loworder problems can be used to create an effective and efficient preconditioner for the corresponding high-order problem. A number of numerical experiments are presented, based on an implementation in the finite element library MFEM. A range of challenging three-dimensional problems are used to corroborate the theoretical properties and demonstrate the flexibility and scalability of the method. [ABSTRACT FROM AUTHOR]

In this paper we introduce general transfer operators between high-order and loworder refined finite element spaces that can be used to couple high-order and low-order simulations. Under natural restrictions on the low-order refined space we prove that both the high-to-low-order and low-to-high-order linear mappings are conservative, constant preserving, and high-order accurate. While the proof holds for affine geometries, numerical experiments indicate that the results hold for more general curved meshes. We present several numerical results confirming our analysis and demonstrate the utility of the new mappings in the context of adaptive mesh refinement and conservative multidiscretization coupling. [ABSTRACT FROM AUTHOR]

Efficient exploitation of exascale architectures requires rethinking of the numerical algorithms used in many large-scale applications. These architectures favor algorithms that expose ultra fine-grain parallelism and maximize the ratio of floating point operations to energy intensive data movement. One of the few viable approaches to achieve high efficiency in the area of PDE discretizations on unstructured grids is to use matrix-free/partially assembled high-order finite element methods, since these methods can increase the accuracy and/or lower the computational time due to reduced data motion. In this paper we provide an overview of the research and development activities in the Center for Efficient Exascale Discretizations (CEED), a co-design center in the Exascale Computing Project that is focused on the development of next-generation discretization software and algorithms to enable a wide range of finite element applications to run efficiently on future hardware. CEED is a research partnership involving more than 30 computational scientists from two US national labs and five universities, including members of the Nek5000, MFEM, MAGMA and PETSc projects. We discuss the CEED co-design activities based on targeted benchmarks, miniapps and discretization libraries and our work on performance optimizations for large-scale GPU architectures. We also provide a broad overview of research and development activities in areas such as unstructured adaptive mesh refinement algorithms, matrix-free linear solvers, high-order data visualization, and list examples of collaborations with several ECP and external applications. [ABSTRACT FROM AUTHOR]

The greater arithmetic intensity of high‐order finite element discretizations makes them attractive for implementation on next‐generation hardware, but assembly of high‐order finite element operators as matrices is prohibitively expensive. As a result, the development of general algebraic solvers for such operators has been an open research challenge. Fast matrix‐free application of high‐order operators has received significant attention in the literature in the context of Poisson‐type problems, but preconditioners and solvers for inverting more general operators are not very well‐developed. In this paper, we consider the problem of preconditioning a definite Maxwell operator at high polynomial order without assembling a matrix. We show that given efficient preconditioners for high‐order H1 finite element problems on the same mesh, efficient H(curl) preconditioners can be constructed in an auxiliary space framework. We demonstrate the resulting preconditioners in a practical setting with tensor‐product basis functions on an unstructured mesh of quadrilaterals. Our approach uses a sparsified H1 solver constructed on a low‐order mesh of the nodal points of the underlying high‐order space, and we show that the resulting H(curl) preconditioner is effective at very high polynomial orders for two‐dimensional model problems with complicated geometry, varying piecewise constant coefficients, and curved elements. The resulting preconditioner scales with nearly optimal O(pd + 1) floating point operation count and optimal O(pd) memory transfer requirements, outperforming existing Maxwell preconditioners in the high‐order regime. [ABSTRACT FROM AUTHOR]

We describe a framework for controlling and improving the quality of high-order finite element meshes based on extensions of the Target-Matrix Optimization Paradigm (TMOP) of [P. Knupp, Eng. Comput., 28 (2012), pp. 419--429]. This approach allows high-order applications to have a very precise control over local mesh quality, while still improving the mesh globally. We address the adaption of various TMOP components to the settings of general isoparametric element mappings, including the mesh quality metric in 2D and 3D, the selection of sample points and the solution of the resulting mesh optimization problem. We also investigate additional practical concerns, such as tangential relaxation and restricting the deviation from the original mesh. The benefits of the new high-order TMOP algorithms are illustrated on a number of test problems and examples from a high-order arbitrary Lagrangian--Eulerian (ALE) application [BLAST: High-order curvilinear finite elements for shock hydrodynamics. [ABSTRACT FROM AUTHOR]

We propose a method that morphs high-order meshes such that their boundaries and interfaces coincide/align with implicitly defined geometries. Our focus is particularly on the case when the target surface is prescribed as the zero isocontour of a smooth discrete function. Common examples of this scenario include using level set functions to represent material interfaces in multimaterial configurations, and evolving geometries in shape and topology optimization. The proposed method formulates the mesh optimization problem as a variational minimization of the sum of a chosen mesh-quality metric using the Target-Matrix Optimization Paradigm (TMOP) and a penalty term that weakly forces the selected faces of the mesh to align with the target surface. The distinct features of the method are use of a source mesh to represent the level set function with sufficient accuracy, and adaptive strategies for setting the penalization weight and selecting the faces of the mesh to be fit to the target isocontour of the level set field. We demonstrate that the proposed method is robust for generating boundary- and interface-fitted meshes for curvilinear domains using different element types in 2D and 3D. [Display omitted] • Implicit high-order meshing using boundary and interface fitting. • Approach targets applications where the target surface is prescribed implicitly using level-set functions. • r-adaptivity method is demonstrated to be robust at adapting easy-to-generate meshes to curvilinear boundaries and interfaces. [ABSTRACT FROM AUTHOR]

In this paper we present a family of scalable preconditioners for matrices arising in the discretization of H(div) problems using the lowest order Raviart--Thomas finite elements. Our approach belongs to the class of "auxiliary space"--based methods and requires only the finite element stiffness matrix plus some minimal additional discretization information about the topology and orientation of mesh entities. We provide a detailed algebraic description of the theory, parallel implementation, and different variants of this parallel auxiliary space divergence solver (ADS) and discuss its relations to the Hiptmair--Xu (HX) auxiliary space decomposition of H(div) [SIAM J. Numer. Anal., 45 (2007), pp. 2483-2509] and to the auxiliary space Maxwell solver AMS [J. Comput. Math., 27 (2009), pp. 604-623]. An extensive set of numerical experiments demonstrates the robustness and scalability of our implementation on large-scale H(div) problems with large jumps in the material coefficients. [ABSTRACT FROM AUTHOR]

The numerical approximation of the Euler equations of gas dynamics in a moving Lagrangian frame is at the heart of many multiphysics simulation algorithms. In this paper, we present a general framework for high-order Lagrangian discretization of these compressible shock hydrodynamics equations using curvilinear finite elements. This method is an extension of the approach outlined in [Dobrev et al., Internat. J. Numer. Methods Fluids, 65 (2010), pp. 1295-1310] and can be formulated for any finite dimensional approximation of the kinematic and thermody-namic fields, including generic finite elements on two- and three-dimensional meshes with triangular, quadrilateral, tetrahedral, or hexahedral zones. We discretize the kinematic variables of position and velocity using a continuous high-order basis function expansion of arbitrary polynomial degree which is obtained via a corresponding high-order parametric mapping from a standard reference element. This enables the use of curvilinear zone geometry, higher-order approximations for fields within a zone, and a pointwise definition of mass conservation which we refer to as strong mass conservation. We discretize the internal energy using a piecewise discontinuous high-order basis function expansion which is also of arbitrary polynomial degree. This facilitates multimaterial hydrodynamics by treating material properties, such as equations of state and constitutive models, as piecewise discontinuous functions which vary within a zone. To satisfy the Rankine-Hugoniot jump conditions at a shock boundary and generate the appropriate entropy, we introduce a general tensor artificial viscosity which takes advantage of the high-order kinematic and thermodynamic information available in each zone. Finally, we apply a generic high-order time discretization process to the semidiscrete equations to develop the fully discrete numerical algorithm. Our method can be viewed as the high-order generalization of the so-called staggered-grid hydrodynamics (SGH) approach and we show that under specific low-order assumptions, we exactly recover the classical SGH method. We present numerical results from an extensive series of verification tests that demonstrate several important practical advantages of using high-order finite elements in this context. [ABSTRACT FROM AUTHOR]

We consider sparse linear systems, where a set of "interior" degrees of freedom have been eliminated in order to reduce the problem size. This elimination is assumed to be local, so the "interior" principal submatrix is block-diagonal, and the resulting Schur complement is still sparse. For it to be beneficial, the elimination process should lead to reduced memory requirements, but equally importantly, it should also result in an algebraic problem that can be solved efficiently. In this paper we propose a general element reduction approach and show how the elimination process can exploit a particular "subzonal" discretization to maintain the sparsity of the Schur complement. We also investigate algebraic multigrid (AMG) solution algorithms applied to the reduced problem. and we discuss the influence of the local elimination on solver-related properties of the matrix, such as near-nullspace preservation and the availability of stable subspace decompositions. We focus on BoomerAMG, a parallel variant of classical Ruge-Stüiben AMG. applied to scalar diffusion problems IV. Henson and U. Yang, Appl. Numer. Math., 41 (2002), pp. 155 177], and the auxiliary-space Maxwell solver (AMS) for electromagnetic diffusion applications [T. Kolev and P. Vassilevski, J. Comput. Math., 27 (2009), pp. 604-623]. In the electromagnetic case, we establish algebraically a reduced version of the Hiptmair-Xu decomposition from [R. Hiptmair and J. Xu, SIAM J. Numev. Anal., 45 (2007), pp. 2483-2509] and consider a modification of the reduction process that targets the singular problems arising in simulations with pure void (zero conductivity) regions. For scalar diffusion problems, our particular stencil analysis shows that the reduction has a positive effect on meshes with stretched elements. We present a number of two-dimensional, three-dimensional, and axisymmetric numerical experiments, which demonstrate that the combination of an appropriately chosen local elimination with the use of the BoomerAMG and AMS solvers can lead to significant improvements in the overall solution time. [ABSTRACT FROM AUTHOR]

In this paper we review a number of auxiliary space based preconditioners for the second order definite and semi-definite Maxwell problems discretized with the lowest order Nédélec finite elements. We discuss the parallel implementation of the most promising of these methods, the ones derived from the recent Hiptmair-Xu (HX) auxiliary space decomposition [Hiptmair and Xu, SIAM J. Numer. Anal., 45 (2007), pp. 2483-2509]. An extensive set of numerical experiments demonstrate the scalability of our implementation on large-scale H(curl) problems. [ABSTRACT FROM AUTHOR]

This paper analyses a two-level preconditioning scheme for H(curl) bilinear forms. The scheme utilizes an auxiliary problem on a related mesh that is more amenable for constructing optimal order multigrid methods. More specifically, we analyse the case when the auxiliary mesh only approximately covers the original domain. The latter assumption is important since it allows for easy construction of nested multilevel spaces on regular auxiliary meshes. Numerical experiments in both two and three space dimensions illustrate the optimal performance of the method. Published in 2007 by John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

This paper studies AMG (algebraic multigrid) methods that utilize energy minimization construction of the interpolation matrices locally, in the setting of element agglomeration AMG. The coarsening in element agglomeration AMG is done by agglomerating fine-grid elements, with coarse element matrices defined by a local Galerkin procedure applied to the matrix assembled from the individual fine-grid element matrices. This local Galerkin procedure involves only the coarse basis restricted to the agglomerated element. To construct the coarse basis, one exploits previously proposed constraint energy minimization procedures now applied to the local matrix. The constraints are that a given set of vectors should be interpolated exactly, not only globally, but also locally on every agglomerated element. The paper provides algorithmic details, as well as a convergence result based on a ‘local-to-global’ energy bound of the resulting multiple-vector fitting AMG interpolation mappings. A particular implementation of the method is illustrated with a set of numerical experiments. Copyright © 2006 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

The MFEM (Modular Finite Element Methods) library is a high-performance C++ library for finite element discretizations. MFEM supports numerous types of finite element methods and is the discretization engine powering many computational physics and engineering applications across a number of domains. This paper describes some of the recent research and development in MFEM, focusing on performance portability across leadership-class supercomputing facilities, including exascale supercomputers, as well as new capabilities and functionality, enabling a wider range of applications. Much of this work was undertaken as part of the Department of Energy's Exascale Computing Project (ECP) in collaboration with the Center for Efficient Exascale Discretizations (CEED). [ABSTRACT FROM AUTHOR]

MAGMA (Matrix Algebra for GPU and Multicore Architectures) is a pivotal open-source library in the landscape of GPU-enabled dense and sparse linear algebra computations. With a repertoire of approximately 750 numerical routines across four precisions, MAGMA is deeply ingrained in the DOE software stack, playing a crucial role in high-performance computing. Notable projects such as ExaConstit, HiOP, MARBL, and STRUMPACK, among others, directly harness the capabilities of MAGMA. In addition, the MAGMA development team has been acknowledged multiple times for contributing to the vendors' numerical software stacks. Looking back over the time of the Exascale Computing Project (ECP), we highlight how MAGMA has adapted to recent changes in modern HPC systems, especially the growing gap between CPU and GPU compute capabilities, as well as the introduction of low precision arithmetic in modern GPUs. We also describe MAGMA's direct impact on several ECP projects. Maintaining portable performance across NVIDIA and AMD GPUs, and with current efforts toward supporting Intel GPUs, MAGMA ensures its adaptability and relevance in the ever-evolving landscape of GPU architectures. [ABSTRACT FROM AUTHOR]

This paper is concerned with the pure displacement problem of planar linear elasticity. Our interest is focused on the locking-free FEM approximation of the problem in the case of almost incompressible material. Crouzeix-Raviart linear finite elements are implemented. An optimal order pure algebraic multiplicative two-level preconditioner for the related stiffness matrix is developed. The proposed construction is based on a proper hierarchical basis of the FEM space. It is important to note that the nodal FEM spaces corresponding to successive levels of mesh refinements are not nested for the non-conforming elements under consideration. Local spectral analysis is applied to determine the scaling parameter of the preconditioner as well as to estimate the related constant in the strengthened CBS inequality. The derived estimates are uniform with respect to the Poisson ratio ν ∈ (0, 0.5). A set of numerical tests is presented to illustrate the accuracy of the FEM approximation, and the convergence rate of the two-level PCG method. Copyright © 1999 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

We propose a general algorithm for nonconforming adaptive mesh refiement (AMR) of unstructured meshes in high-order finite element codes. Our focus is on h-refinement with a fixed polynomial order. The algorithm handles triangular, quadrilateral, hexahedral, and prismatic meshes of arbitrarily high-order curvature, for any order finite element space in the de Rham sequence. We present a exible data structure for meshes with hanging nodes and a general procedure to construct the conforming interpolation operator, both in serial and in parallel. The algorithm and data structure allow anisotropic refinement of tensor product elements in two and three dimensions and support unlimited refinement ratios of adjacent elements. We report numerical experiments verifying the correctness of the algorithms and perform a parallel scaling study to show that we can adapt meshes containing billions of elements and run effciently on 393,000 parallel tasks. Finally, we illustrate the integration of dynamic AMR into a high-order Lagrangian hydrodynamics solver. [ABSTRACT FROM AUTHOR]

We present a new approach for multi-material arbitrary Lagrangian-Eulerian (ALE) hydrodynamics simulations based on high-order finite elements posed on high-order curvilinear meshes. The method builds on and extends our previous work in the Lagrangian [V. A. Dobrev, T. V. Kolev, and R. N. Rieben, SIAM J. Sci. Comput., 34 (2012), pp. B606-B641] and remap [R. W. Anderson et al., Internat. J. Numer. Methods Fluids, 77 (2015), pp. 249-273] phases of ALE, and depends critically on a functional perspective that enables subzonal physics and material modeling [V. A. Dobrev et al., Internat. J. Numer. Methods Fluids, 82 (2016), pp. 689-706]. Curvilinear mesh relaxation is based on node movement, which is determined through the solution of an elliptic equation. The remap phase is posed in terms of advecting state variables between two meshes over a fictitious time interval. The resulting advection equation is solved by a discontinuous Galerkin (DG) formulation, combined with a customized Flux Corrected Transport (FCT) type algorithm. Because conservative fields are remapped, additional synchronization steps are introduced to preserve bounds with respect to primal fields. These steps include modification of the low-order FCT solutions, def- inition of conservative FCT uxes based on primal field bounds, and monotone transitions between primal and conservative fields. This paper describes the mathematical formulation and properties of our approach and reports a number of numerical results from its implementation in the BLAST code [BLAST: High-order finite element Lagrangian hydrocode, http://www.llnl.gov/CASC/blast]. Additional details can be found in [R. W. Anderson et al., High-Order Multi-Material ALE Hy- drodynamics (Extended Version), Tech. report LLNL-JRNL-706339, Lawrence Livermore National Laboratory, Livermore, CA, 2016]. [ABSTRACT FROM AUTHOR]

With the introduction of advanced heterogeneous computing architectures based on GPU accelerators, large-scale production codes have had to rethink their numerical algorithms and incorporate new programming models and memory management strategies in order to run efficiently on the latest supercomputers. In this work we discuss our co-design strategy to address these challenges and achieve performance and portability with MARBL, a next-generation multi-physics code in development at Lawrence Livermore National Laboratory. We present a two-fold approach, wherein new hardware is used to motivate both new algorithms and new abstraction layers, resulting in a single source application code suitable for a variety of platforms. Focusing on MARBL's ALE hydrodynamics package, we demonstrate scalability on different platforms and highlight that many of our innovations have been contributed back to open-source software libraries, such as MFEM (finite element algorithms) and RAJA (kernel abstractions). [ABSTRACT FROM AUTHOR]

The magnetohydrodynamics (MHD) equations are continuum models used in the study of a wide range of plasma physics systems, including the evolution of complex plasma dynamics in tokamak disruptions. However, efficient numerical solution methods for MHD are extremely challenging due to disparate time and length scales, strong hyperbolic phenomena, and nonlinearity. Therefore the development of scalable, implicit MHD algorithms and high-resolution adaptive mesh refinement strategies is of considerable importance. In this work, we develop a high-order stabilized finite-element algorithm for the reduced visco-resistive MHD equations based on the MFEM finite element library (mfem.org). The scheme is fully implicit, solved with the Jacobian-free Newton-Krylov (JFNK) method with a physics-based preconditioning strategy. Our preconditioning strategy is a generalization of the physics-based preconditioning methods in Chacón et al. (2002) [3] to adaptive, stabilized finite elements. Algebraic multigrid methods are used to invert sub-block operators to achieve scalability. A parallel adaptive mesh refinement scheme with dynamic load-balancing is implemented to efficiently resolve the multi-scale spatial features of the system. Our implementation uses the MFEM framework, which provides arbitrary-order polynomials and flexible adaptive conforming and non-conforming meshes capabilities. Results demonstrate the accuracy, efficiency, and scalability of the implicit scheme in the presence of large scale disparity. The potential of the AMR approach is demonstrated on an island coalescence problem in the high Lundquist-number regime (≥ 10 7) with the successful resolution of plasmoid instabilities and thin current sheets. • Fully implicit SUPG-based stabilized finite elements for resistive MHD. • Scalable physics-based preconditioning. • A parallel AMR algorithm with dynamic load balancing in MFEM. • Demonstration of excellent parallel and algorithmic scaling up to 4096 CPUs. • Multi-scale simulations for plasmoid instabilities of high Lundquist number. [ABSTRACT FROM AUTHOR]

This paper investigates the properties of smoothers in the context of algebraic multigrid (AMG) running oil parallel computers with potentially millions of processors. The development of multigrid smoothers in this case is challenging, because some of the best relaxation schemes, such as the Gauss-Seidel (GS) algorithm, are inherently sequential. Based on the sharp two-grid multigrid theory from [R. D. Falgout and P. S. Vassilevski, SIAM J. Numer. Anal., 42 (2004), pp. 1669-1693] and [R. D. Falgout, P. S. Vassilevski, and L. T. Zikatanov, Numer. Linear Algebra Appl., 12 (2005). pp. 471-494] we characterize the smoothing properties of a number of practical candidates for parallel smoothers, including several C-F, polynomial, and hybrid schemes. We show, in particular, that the popular hybrid GS algorithm has multigrid smoothing properties which are independent of the number of processors in many practical applications, provided that the problem size per processor is large enough. This is encouraging news for the scalability of AMG on ultraparallel computers. We also introduce the more robust l1 smoothers, which are always convergent and have already proven essential for the parallel solution of some electromagnetic problems [T. Kolev and P. Vassilevski, J. Comput. Math., 27 (2009), pp. 604-623]. [ABSTRACT FROM AUTHOR]

This work develops novel time integration methods for the compressible Euler equations in the Lagrangian frame that are of arbitrary high order and exactly preserve the mass, momentum, and total energy of the system. The equations are considered in nonconservative form, that is, common for staggered grid hydrodynamics (SGH) methods; namely, the evolved quantities are mass, momentum, and internal energy. A general family of time integration schemes is formulated, and practical pairs for orders three and four are derived. Numerical results on standard hydrodynamics benchmarks confirm the high-order convergence on smooth problems and the exact numerical preservation of all physically conserved quantities. [ABSTRACT FROM AUTHOR]

MFEM is an open-source, lightweight, flexible and scalable C++ library for modular finite element methods that features arbitrary high-order finite element meshes and spaces, support for a wide variety of discretization approaches and emphasis on usability, portability, and high-performance computing efficiency. MFEM's goal is to provide application scientists with access to cutting-edge algorithms for high-order finite element meshing, discretizations and linear solvers, while enabling researchers to quickly and easily develop and test new algorithms in very general, fully unstructured, high-order, parallel and GPU-accelerated settings. In this paper we describe the underlying algorithms and finite element abstractions provided by MFEM, discuss the software implementation, and illustrate various applications of the library. [ABSTRACT FROM AUTHOR]

In this work, we introduce a new residual distribution (RD) framework for the design of bound-preserving high-resolution finite element schemes. The continuous and discontinuous Galerkin discretizations of the linear advection equation are modified to construct local extremum diminishing (LED) approximations. To that end, we perform mass lumping and redistribute the element residuals in a manner which guarantees the LED property. The hierarchical correction procedure for high-order Bernstein finite element discretizations involves localization to subcells and definition of bound-preserving weights for subcell contributions. Using strong stability preserving (SSP) Runge–Kutta methods for time integration, we prove the validity of discrete maximum principles under CFL-like time step restrictions. The low-order version of our method has roughly the same accuracy as the one derived from a piecewise (multi)-linear approximation on a submesh with the same nodal points. In high-order extensions, we use an element-based flux-corrected transport (FCT) algorithm which can be interpreted as a nonlinear RD scheme. The proposed LED corrections are tailor-made for matrix-free implementations which avoid the rapidly growing cost of matrix assembly for high-order Bernstein elements. The results for 1D, 2D, and 3D test problems compare favorably to those obtained with the best matrix-based approaches. • High-order Bernstein finite element discretizations of linear advection problems. • Matrix-free element-level corrections of continuous Galerkin and DG schemes. • Local extremum diminishing property under realistic time step restrictions. • p -independent convergence behavior of nonlinear low-order approximations. • Bound-preserving antidiffusive corrections based on localized FCT algorithms. [ABSTRACT FROM AUTHOR]

Performance tests and analyses are critical to effective high-performance computing software development and are central components in the design and implementation of computational algorithms for achieving faster simulations on existing and future computing architectures for large-scale application problems. In this article, we explore performance and space-time trade-offs for important compute-intensive kernels of large-scale numerical solvers for partial differential equations (PDEs) that govern a wide range of physical applications. We consider a sequence of PDE-motivated bake-off problems designed to establish best practices for efficient high-order simulations across a variety of codes and platforms. We measure peak performance (degrees of freedom per second) on a fixed number of nodes and identify effective code optimization strategies for each architecture. In addition to peak performance, we identify the minimum time to solution at 80% parallel efficiency. The performance analysis is based on spectral and p -type finite elements but is equally applicable to a broad spectrum of numerical PDE discretizations, including finite difference, finite volume, and h -type finite elements. [ABSTRACT FROM AUTHOR]

With current trends in computer architectures leading towards systems with more, but not faster, processors, faster time-to-solution must come from greater parallelism. We present a family of truly multilevel approaches to parallel time integration based on multigrid reduction (MGR) principles. The resulting multigrid-reduction-in-time (MGRIT) algorithms are non-intrusive approaches, which directly use an existing time propagator and, thus, can easily exploit substantially more computational resources then standard sequential time-stepping. Furthermore, we demonstrate that MGRIT offers excellent strong and weak parallel scaling up to thousands of processors for solving diffusion equations in two and three space dimensions. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim) [ABSTRACT FROM AUTHOR]

In this paper we present a new GPU-oriented mesh optimization method based on high-order finite elements. Our approach relies on node movement with fixed topology, through the Target-Matrix Optimization Paradigm (TMOP) and uses a global nonlinear solve over the whole computational mesh, i.e., all mesh nodes are moved together. A key property of the method is that the mesh optimization process is recast in terms of finite element operations, which allows us to utilize recent advances in the field of GPU-accelerated high-order finite element algorithms. For example, we reduce data motion by using tensor factorization and matrix-free methods, which have superior performance characteristics compared to traditional full finite element matrix assembly and offer advantages for GPU-based HPC hardware. We describe the major mathematical components of the method along with their efficient GPU-oriented implementation. In addition, we propose an easily reproducible mesh optimization test that can serve as a performance benchmark for the mesh optimization community. • Introduces a GPU-oriented mesh optimization method based on high-order finite elements. • Reduction of data motion is achieved by tensor factorization and matrix-free methods. • Proposes a reproducible performance benchmark for mesh optimization. [ABSTRACT FROM AUTHOR]

In this paper we describe the research and development activities in the Center for Efficient Exascale Discretization within the US Exascale Computing Project, targeting state-of-the-art high-order finite-element algorithms for high-order applications on GPU-accelerated platforms. We discuss the GPU developments in several components of the CEED software stack, including the libCEED, MAGMA, MFEM, libParanumal, and Nek projects. We report performance and capability improvements in several CEED-enabled applications on both NVIDIA and AMD GPU systems. • Exascale machines require rethinking of the large-scale simulation codes algorithms. • High-order finite elements can enhance the performance of exascale applications. • The CEED project in ECP develops high-order algorithms optimized for GPU platforms. • Several CEED components have been ported to both NVIDIA and AMD GPU systems. [ABSTRACT FROM AUTHOR]