Matrix-free subcell residual distribution for Bernstein finite elements: Monolithic limiting

Hajduk, Hennes; Kuzmin, Dmitri; Kolev, Tzanio; Tomov, Vladimir; Tomaš, Ignacio; Shadid, John N.

doi:10.1016/j.compfluid.2020.104451

Cited by 17 publications

(29 citation statements)

References 23 publications

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“…Remhos solves the pure advection equations that are used to perform conservative and monotonic DG advection-based discontinuous field interpolation, or “remap” (Remhos). Remhos combines discretization methods described in the following articles: (Anderson et al, 2015, 2017, 2018; Hajduk et al, 2020a, 2020b). It exposes the principal computational kernels of explicit time-dependent Discontinuous Galerkin advection methods, including monotonicity treatment computations that are characteristic to FCT (Flux Corrected Transport) methods.…”

Section: Miniappsmentioning

confidence: 99%

Efficient exascale discretizations: High-order finite element methods

Kolev

Fischer

Min

et al. 2021

The International Journal of High Performance Computing Applica

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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.

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Section: Miniappsmentioning

confidence: 99%

Efficient exascale discretizations: High-order finite element methods

Kolev

Fischer

Min

et al. 2021

The International Journal of High Performance Computing Applica

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“…This approach does not guarantee exact sparsity for general velocity fields v(x). As a consequence, the resulting schemes become less accurate as the polynomial degree p is increased while keeping the total number of DoFs N h fixed [22].…”

Section: Low-order Bernstein Finite Element Discretizationmentioning

confidence: 99%

“…As expected, the approximation calculated with the full stencil Q 2 scheme proves more dissipative than the compact-stencil Q 1 and Q 2 approximations. In contrast to the subcell upwinding strategy employed in [22,41], the low-order scheme defined by (15) preserves the Q 1 sparsity pattern exactly even for nonuniform velocity fields and nonlinear flux functions. This remarkable property eliminates a major bottleneck to achieving high performance and p-independent convergence behavior with matrix-based algebraic flux correction schemes.…”

Section: Solid Body Rotationmentioning

confidence: 99%

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Subcell flux limiting for high-order Bernstein finite element discretizations of scalar hyperbolic conservation laws

Kuzmin

Luna

2020

Journal of Computational Physics

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This work extends the concepts of algebraic flux correction and convex limiting to continuous high-order Bernstein finite element discretizations of scalar hyperbolic problems. Using an array of adjustable diffusive fluxes, the standard Galerkin approximation is transformed into a nonlinear high-resolution scheme which has the compact sparsity pattern of the piecewise-linear or multilinear subcell discretization. The representation of this scheme in terms of invariant domain preserving states makes it possible to prove the validity of local discrete maximum principles under CFL-like conditions. In contrast to predictor-corrector approaches based on the flux-corrected transport methodology, the proposed flux limiting strategy is monolithic; i.e., limited antidiffusive terms are incorporated into the well-defined residual of a nonlinear (semi-)discrete problem. A stabilized high-order Galerkin discretization is recovered if no limiting is performed. In the limited version, the compact stencil property prevents direct mass exchange between nodes that are not nearest neighbors. A formal proof of sparsity is provided for simplicial and box elements. The involved element contributions can be calculated efficiently making use of matrix-free algorithms and precomputed element matrices of the reference element. Numerical studies for Q 2 discretizations of linear and nonlinear two-dimensional test problems illustrate the virtues of monolithic convex limiting based on subcell flux decompositions.

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“…Limiting techniques of this kind have also been successfully applied to CG [18,27] and DG [10] discretizations. The use of localized subcell limiters was found to be essential in extensions to high-order Bernstein finite elements [12,11,19,27]. An hp-adaptive approach to subcell limiting was introduced in [20].…”

Section: Introductionmentioning

confidence: 99%

A subcell-enriched Galerkin method for advection problems

Rupp¹,

Moritz²,

Aizinger³

2020

Preprint

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In this work, we introduce a generalization of the enriched Galerkin (EG) method. The key feature of our scheme is an adaptive two-mesh approach that, in addition to the standard enrichment of a conforming finite element discretization via discontinuous degrees of freedom, allows to subdivide selected (e.g. troubled) mesh cells in a non-conforming fashion and to use further discontinuous enrichment on this finer submesh. We prove stability and sharp a priori error estimates for a linear advection equation by using a specially tailored projection and conducting some parts of a standard convergence analysis for both meshes. By allowing an arbitrary degree of enrichment on both, the coarse and the fine mesh (also including the case of no enrichment), our analysis technique is very general in the sense that our results cover the range from the standard continuous finite element method to the standard discontinuous Galerkin (DG) method with (or without) local subcell enrichment. Numerical experiments confirm our analytical results and indicate good robustness of the proposed method.

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Matrix-free subcell residual distribution for Bernstein finite elements: Monolithic limiting

Cited by 17 publications

References 23 publications

Efficient exascale discretizations: High-order finite element methods

Efficient exascale discretizations: High-order finite element methods

Subcell flux limiting for high-order Bernstein finite element discretizations of scalar hyperbolic conservation laws

A subcell-enriched Galerkin method for advection problems

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