TY - GEN

T1 - Discrete energy laws for the first-order system least-squares finite-element approach

AU - Adler, J. H.

AU - Lashuk, I.

AU - MacLachlan, S. P.

AU - Zikatanov, L. T.

N1 - Funding Information:
Acknowledgements. The work of J. H. Adler was supported in part by NSF DMS-1216972. I. V. Lashuk was supported in part by NSF DMS-1216972 (Tufts University) and DMS-1418843 (Penn State). S. P. MacLachlan was partially supported by an NSERC Discovery Grant. The research of L. T. Zikatanov was supported in part by NSF DMS-1720114 and the Department of Mathematics at Tufts University.
Publisher Copyright:
© Springer International Publishing AG 2018.

PY - 2018

Y1 - 2018

N2 - This paper analyzes the discrete energy laws associated with first-order system least-squares (FOSLS) discretizations of time-dependent partial differential equations. Using the heat equation and the time-dependent Stokes’ equation as examples, we discuss how accurately a FOSLS finite-element formulation adheres to the underlying energy law associated with the physical system. Using regularity arguments involving the initial condition of the system, we are able to give bounds on the convergence of the discrete energy law to its expected value (zero in the examples presented here). Numerical experiments are performed, showing that the discrete energy laws hold with order O(h2 p), where h is the mesh spacing and p is the order of the finite-element space. Thus, the energy law conformance is held with a higher order than the expected, O(hp), convergence of the finite-element approximation. Finally, we introduce an abstract framework for analyzing the energy laws of general FOSLS discretizations.

AB - This paper analyzes the discrete energy laws associated with first-order system least-squares (FOSLS) discretizations of time-dependent partial differential equations. Using the heat equation and the time-dependent Stokes’ equation as examples, we discuss how accurately a FOSLS finite-element formulation adheres to the underlying energy law associated with the physical system. Using regularity arguments involving the initial condition of the system, we are able to give bounds on the convergence of the discrete energy law to its expected value (zero in the examples presented here). Numerical experiments are performed, showing that the discrete energy laws hold with order O(h2 p), where h is the mesh spacing and p is the order of the finite-element space. Thus, the energy law conformance is held with a higher order than the expected, O(hp), convergence of the finite-element approximation. Finally, we introduce an abstract framework for analyzing the energy laws of general FOSLS discretizations.

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U2 - 10.1007/978-3-319-73441-5_1

DO - 10.1007/978-3-319-73441-5_1

M3 - Conference contribution

AN - SCOPUS:85041739848

SN - 9783319734408

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 3

EP - 20

BT - Large-Scale Scientific Computing - 11th International Conference, LSSC 2017, Revised Selected Papers

A2 - Lirkov, Ivan

A2 - Margenov, Svetozar

PB - Springer Verlag

T2 - 11th International Conference on Large-Scale Scientific Computations, LSSC 2017

Y2 - 11 September 2017 through 15 September 2017

ER -