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Volume 1, Issue 1
A priori Error Analysis of a Discontinuous Galerkin Method for Cahn–Hilliard–Navier–Stokes Equations

Chen Liu & Béatrice Rivière

DOI: 10.4208/csiam-am.2020-0005

CSIAM Trans. Appl. Math., 1 (2020), pp. 104-141.

Published online: 2020-03

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  • Abstract

In this paper, we analyze an interior penalty discontinuous Galerkin method for solving the coupled Cahn–Hilliard and Navier–Stokes equations. We prove unconditional unique solvability of the discrete system, and we derive stability bounds without any restrictions on the chemical energy density function. The numerical solutions satisfy a discrete energy dissipation law and mass conservation laws. Convergence of the method is obtained by obtaining optimal a priori error estimates.

  • Keywords

Cahn–Hilliard–Navier–Stokes, interior penalty discontinuous Galerkin method, existence, uniqueness, stability, error estimates.

  • AMS Subject Headings

35G25, 65M60, 65M12, 76D05

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

chen.liu@rice.edu (Chen Liu)

riviere@rice.edu (Béatrice Rivière)

  • BibTex
  • RIS
  • TXT
@Article{CSIAM-AM-1-104, author = {Liu , Chen and Rivière , Béatrice}, title = {A priori Error Analysis of a Discontinuous Galerkin Method for Cahn–Hilliard–Navier–Stokes Equations}, journal = {CSIAM Transactions on Applied Mathematics}, year = {2020}, volume = {1}, number = {1}, pages = {104--141}, abstract = {

In this paper, we analyze an interior penalty discontinuous Galerkin method for solving the coupled Cahn–Hilliard and Navier–Stokes equations. We prove unconditional unique solvability of the discrete system, and we derive stability bounds without any restrictions on the chemical energy density function. The numerical solutions satisfy a discrete energy dissipation law and mass conservation laws. Convergence of the method is obtained by obtaining optimal a priori error estimates.

}, issn = {2708-0579}, doi = {https://doi.org/10.4208/csiam-am.2020-0005}, url = {http://global-sci.org/intro/article_detail/csiam-am/16795.html} }
TY - JOUR T1 - A priori Error Analysis of a Discontinuous Galerkin Method for Cahn–Hilliard–Navier–Stokes Equations AU - Liu , Chen AU - Rivière , Béatrice JO - CSIAM Transactions on Applied Mathematics VL - 1 SP - 104 EP - 141 PY - 2020 DA - 2020/03 SN - 1 DO - http://doi.org/10.4208/csiam-am.2020-0005 UR - https://global-sci.org/intro/article_detail/csiam-am/16795.html KW - Cahn–Hilliard–Navier–Stokes, interior penalty discontinuous Galerkin method, existence, uniqueness, stability, error estimates. AB -

In this paper, we analyze an interior penalty discontinuous Galerkin method for solving the coupled Cahn–Hilliard and Navier–Stokes equations. We prove unconditional unique solvability of the discrete system, and we derive stability bounds without any restrictions on the chemical energy density function. The numerical solutions satisfy a discrete energy dissipation law and mass conservation laws. Convergence of the method is obtained by obtaining optimal a priori error estimates.

Liu , Chen and Rivière , Béatrice. (2020). A priori Error Analysis of a Discontinuous Galerkin Method for Cahn–Hilliard–Navier–Stokes Equations. CSIAM Transactions on Applied Mathematics. 1 (1). 104-141. doi:10.4208/csiam-am.2020-0005
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