Optimization-based, property-preserving finite element methods for scalar advection equations and their connection to Algebraic Flux Correction

Bochev, Pavel; Ridzal, Denis; D’Elia, Marta; Perego, Mauro; Peterson, Kara

doi:10.1016/j.cma.2020.112982

Optimization-based, property-preserving finite element methods for scalar advection equations and their connection to Algebraic Flux Correction

Journal Article · Thu May 28 00:00:00 EDT 2020 · Computer Methods in Applied Mechanics and Engineering

DOI:https://doi.org/10.1016/j.cma.2020.112982· OSTI ID:1644056

Bochev, Pavel ^[1]; Ridzal, Denis ^[1]; D’Elia, Marta ^[1]; Perego, Mauro ^[1]; Peterson, Kara ^[1]

Sandia National Lab. (SNL-NM), Albuquerque, NM (United States). Center for Computing Research

In this paper, we continue our efforts to exploit optimization and control ideas as a common foundation for the development of property-preserving numerical methods. Here we focus on a class of scalar advection equations whose solutions have fixed mass in a given Eulerian region and constant bounds in any Lagrangian volume. Our approach separates discretization of the equations from the preservation of their solution properties by treating the latter as optimization constraints. This relieves the discretization process from having to comply with additional restrictions and makes stability and accuracy the sole considerations in its design. A property-preserving solution is then sought as a state that minimizes the distance to an optimally accurate but not property-preserving target solution computed by the scheme, subject to constraints enforcing discrete proxies of the desired properties. Furthermore, we consider two such formulations in which the optimization variables are given by the nodal solution values and suitably defined nodal fluxes, respectively. A key result of the paper reveals that a standard Algebraic Flux Correction (AFC) scheme is a modified version of the second formulation obtained by shrinking its feasible set to a hypercube. In conclusion, we present numerical studies illustrating the optimization-based formulations and comparing them with AFC

View Accepted Manuscript (DOE)

Research Organization:: Sandia National Laboratories (SNL-NM), Albuquerque, NM (United States); Sandia National Laboratories, Livermore, CA

Sponsoring Organization:: USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR); USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR) (SC-21)

Grant/Contract Number:: AC04-94AL85000

OSTI ID:: 1644056

Alternate ID(s):: OSTI ID: 1776277

Report Number(s):: SAND--2019-6097J; 676176

Journal Information:: Computer Methods in Applied Mechanics and Engineering, Journal Name: Computer Methods in Applied Mechanics and Engineering Vol. 367; ISSN 0045-7825

Publisher:: ElsevierCopyright Statement

Country of Publication:: United States

Language:: English

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Optimization-based, property-preserving finite element methods for scalar advection equations and their connection to Algebraic Flux Correction

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