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Improving Conservation for First-Order System Least-Squares Finite-Element Methods

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Numerical Solution of Partial Differential Equations: Theory, Algorithms, and Their Applications

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 45))

Abstract

The first-order system least-squares (FOSLS) finite element method for solving partial differential equations has many advantages, including the construction of symmetric positive definite algebraic linear systems that can be solved efficiently with multilevel iterative solvers. However, one drawback of the method is the potential lack of conservation of certain properties. One such property is conservation of mass. This paper describes a strategy for achieving mass conservation for a FOSLS system by changing the minimization process to that of a constrained minimization problem. If the space of corresponding Lagrange multipliers contains the piecewise constants, then local mass conservation is achieved similarly to the standard mixed finite-element method. To make the strategy more robust and not add too much computational overhead to solving the resulting saddle-point system, an overlapping Schwarz process is used.

Mathematics Subject Classification (2010): 65F10, 65N20, 65N30

The work of the author “P.S. Vassilevski” was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344

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Author information

Authors and Affiliations

  1. Department of Mathematics, Tufts University, Medford, MA, 02155, USA

    J. H. Adler

  2. Center for Applied Scientific Computing, Lawrence Livermore National Laboratory, 808, L-560, Livermore, CA, 94551, USA

    P. S. Vassilevski

Authors
  1. J. H. Adler
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  2. P. S. Vassilevski
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Correspondence to P. S. Vassilevski .

Editor information

Editors and Affiliations

  1. , Dept. of Flow and Material Simulation, Fraunhofer Institute for Industrial Math, Fraunhofer-Platz 1, Kaiserslautern, 67663, Germany

    Oleg P. Iliev

  2. Institute for Parallel Processing, Bulgarian Academy of Sciences, Acad. G. Bonchev Str. 25A, Sofia, 1113, Bulgaria

    Svetozar D. Margenov

  3. , Dept. of Mathematical and Statistical Sc, University of Alberta, 677 Central Academic Building, Edmonton, T6G 2G1, Alberta, Canada

    Peter D Minev

  4. Center for Applied Scientific Computing, Lawrence Livermore National Laboratory, East Avenue 7000, Livermore, 94550, California, USA

    Panayot S. Vassilevski

  5. , Department of Mathematics, Pennsylvania State University, 310 McAllister Building, University Park, 16802, Pennsylvania, USA

    Ludmil T Zikatanov

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Adler, J.H., Vassilevski, P.S. (2013). Improving Conservation for First-Order System Least-Squares Finite-Element Methods. In: Iliev, O., Margenov, S., Minev, P., Vassilevski, P., Zikatanov, L. (eds) Numerical Solution of Partial Differential Equations: Theory, Algorithms, and Their Applications. Springer Proceedings in Mathematics & Statistics, vol 45. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-7172-1_1

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