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Title: Explicit synchronous partitioned algorithms for interface problems based on Lagrange multipliers

Abstract

Traditional explicit partitioned schemes exchange boundary conditions between subdomains and can be related to iterative solution methods for the coupled problem. As a result, these schemes may require multiple subdomain solves, acceleration techniques, or optimized transmission conditions to achieve sufficient accuracy and/or stability. We present a new synchronous partitioned method derived from a well-posed mixed finite element formulation of the coupled problem. We transform the resulting Differential Algebraic Equation (DAE) to a Hessenberg index-1 form in which the algebraic equation defines the Lagrange multiplier as an implicit function of the states. Using this fact we eliminate the multiplier and reduce the DAE to a system of explicit ODEs for the states. Explicit time integration both discretizes this system in time and decouples its equations. As a result, the temporal accuracy and stability of our formulation are governed solely by the accuracy and stability of the explicit scheme employed and are not subject to additional stability considerations as in traditional partitioned schemes. Here, we establish sufficient conditions for the formulation to be well-posed and prove that classical mortar finite elements on the interface are a stable choice for the Lagrange multiplier. We show that in this case the condition number ofmore » the Schur complement involved in the elimination of the multiplier is bounded by a constant. The paper concludes with numerical examples illustrating the approach for two different interface problems.« less

Authors:
 [1];  [1]; ORCiD logo [1]
  1. Sandia National Lab. (SNL-NM), Albuquerque, NM (United States). Center for Computing Research
Publication Date:
Research Org.:
Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
Sponsoring Org.:
USDOE National Nuclear Security Administration (NNSA)
OSTI Identifier:
1492351
Report Number(s):
SAND-2018-14135J
Journal ID: ISSN 0898-1221; 671006
Grant/Contract Number:  
AC04-94AL85000
Resource Type:
Journal Article: Accepted Manuscript
Journal Name:
Computers and Mathematics with Applications (Oxford)
Additional Journal Information:
Journal Name: Computers and Mathematics with Applications (Oxford); Journal ID: ISSN 0898-1221
Publisher:
Elsevier
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING; Transmission problem; Mortar method; Lagrange multiplier; Advection–diffusion; Elastodynamics; Hessenberg index-1 DAE

Citation Formats

Peterson, Kara J., Bochev, Pavel B., and Kuberry, Paul Allen. Explicit synchronous partitioned algorithms for interface problems based on Lagrange multipliers. United States: N. p., 2018. Web. doi:10.1016/j.camwa.2018.09.045.
Peterson, Kara J., Bochev, Pavel B., & Kuberry, Paul Allen. Explicit synchronous partitioned algorithms for interface problems based on Lagrange multipliers. United States. doi:10.1016/j.camwa.2018.09.045.
Peterson, Kara J., Bochev, Pavel B., and Kuberry, Paul Allen. Fri . "Explicit synchronous partitioned algorithms for interface problems based on Lagrange multipliers". United States. doi:10.1016/j.camwa.2018.09.045.
@article{osti_1492351,
title = {Explicit synchronous partitioned algorithms for interface problems based on Lagrange multipliers},
author = {Peterson, Kara J. and Bochev, Pavel B. and Kuberry, Paul Allen},
abstractNote = {Traditional explicit partitioned schemes exchange boundary conditions between subdomains and can be related to iterative solution methods for the coupled problem. As a result, these schemes may require multiple subdomain solves, acceleration techniques, or optimized transmission conditions to achieve sufficient accuracy and/or stability. We present a new synchronous partitioned method derived from a well-posed mixed finite element formulation of the coupled problem. We transform the resulting Differential Algebraic Equation (DAE) to a Hessenberg index-1 form in which the algebraic equation defines the Lagrange multiplier as an implicit function of the states. Using this fact we eliminate the multiplier and reduce the DAE to a system of explicit ODEs for the states. Explicit time integration both discretizes this system in time and decouples its equations. As a result, the temporal accuracy and stability of our formulation are governed solely by the accuracy and stability of the explicit scheme employed and are not subject to additional stability considerations as in traditional partitioned schemes. Here, we establish sufficient conditions for the formulation to be well-posed and prove that classical mortar finite elements on the interface are a stable choice for the Lagrange multiplier. We show that in this case the condition number of the Schur complement involved in the elimination of the multiplier is bounded by a constant. The paper concludes with numerical examples illustrating the approach for two different interface problems.},
doi = {10.1016/j.camwa.2018.09.045},
journal = {Computers and Mathematics with Applications (Oxford)},
issn = {0898-1221},
number = ,
volume = ,
place = {United States},
year = {2018},
month = {10}
}

Journal Article:
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