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Matrix-free subcell residual distribution for Bernstein finite element discretizations of linear advection equations


Hajduk, Hennes; Kuzmin, Dmitri; Kolev, Tzanio; Abgrall, Rémi (2020). Matrix-free subcell residual distribution for Bernstein finite element discretizations of linear advection equations. Computer Methods in Applied Mechanics and Engineering, 359:112658.

Abstract

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.

Abstract

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.

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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:340 Law
610 Medicine & health
510 Mathematics
Scopus Subject Areas:Physical Sciences > Computational Mechanics
Physical Sciences > Mechanics of Materials
Physical Sciences > Mechanical Engineering
Physical Sciences > General Physics and Astronomy
Physical Sciences > Computer Science Applications
Uncontrolled Keywords:Mechanical Engineering, General Physics and Astronomy, Mechanics of Materials, Computational Mechanics, Computer Science Applications
Language:English
Date:1 February 2020
Deposited On:06 May 2020 12:18
Last Modified:29 Jul 2020 15:07
Publisher:Elsevier
ISSN:0045-7825
OA Status:Closed
Publisher DOI:https://doi.org/10.1016/j.cma.2019.112658
Project Information:
  • : FunderSNSF
  • : Grant ID200020_175784
  • : Project TitleSolving advection dominated problems with high order schemes with polygonal meshes: application to compressible and incompressible flow problems

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