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Volume 20, Issue 4
Unconditional Energy Stability and Solvability for a C0 Interior Penalty Method for a Sixth-Order Equation Modeling Microemulsions

Amanda E. Diegel & Natasha S. Sharma

DOI: 10.4208/ijnam2023-1019

Int. J. Numer. Anal. Mod., 20 (2023), pp. 459-477.

Published online: 2023-05

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  • Abstract

We consider a C0 interior penalty finite element approximation of a sixth-order Cahn-Hilliard type equation that models the dynamics of phase transitions in ternary oil-water-surfactant systems. The nonlinear sixth-order parabolic equation is expressed in a mixed form whereby a second-order (in space) parabolic equation and an algebraic fourth-order (in space) nonlinear equation are considered. The temporal discretization is chosen so that a discrete energy law can be established leading to unconditional energy stability. Additionally, we show that the numerical method is unconditionally uniquely solvable. We conclude with several numerical experiments demonstrating the unconditional stability and first-order accuracy of the proposed method.

  • Keywords

Finite element, Cahn-Hilliard, unconditional energy stability, microemulsions and unique solvability.

  • AMS Subject Headings

65M60, 74N20, 74S05

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{IJNAM-20-459, author = {Diegel , Amanda E. and Sharma , Natasha S.}, title = {Unconditional Energy Stability and Solvability for a C0 Interior Penalty Method for a Sixth-Order Equation Modeling Microemulsions }, journal = {International Journal of Numerical Analysis and Modeling}, year = {2023}, volume = {20}, number = {4}, pages = {459--477}, abstract = {

We consider a C0 interior penalty finite element approximation of a sixth-order Cahn-Hilliard type equation that models the dynamics of phase transitions in ternary oil-water-surfactant systems. The nonlinear sixth-order parabolic equation is expressed in a mixed form whereby a second-order (in space) parabolic equation and an algebraic fourth-order (in space) nonlinear equation are considered. The temporal discretization is chosen so that a discrete energy law can be established leading to unconditional energy stability. Additionally, we show that the numerical method is unconditionally uniquely solvable. We conclude with several numerical experiments demonstrating the unconditional stability and first-order accuracy of the proposed method.

}, issn = {2617-8710}, doi = {https://doi.org/10.4208/ijnam2023-1019}, url = {http://global-sci.org/intro/article_detail/ijnam/21711.html} }
TY - JOUR T1 - Unconditional Energy Stability and Solvability for a C0 Interior Penalty Method for a Sixth-Order Equation Modeling Microemulsions AU - Diegel , Amanda E. AU - Sharma , Natasha S. JO - International Journal of Numerical Analysis and Modeling VL - 4 SP - 459 EP - 477 PY - 2023 DA - 2023/05 SN - 20 DO - http://doi.org/10.4208/ijnam2023-1019 UR - https://global-sci.org/intro/article_detail/ijnam/21711.html KW - Finite element, Cahn-Hilliard, unconditional energy stability, microemulsions and unique solvability. AB -

We consider a C0 interior penalty finite element approximation of a sixth-order Cahn-Hilliard type equation that models the dynamics of phase transitions in ternary oil-water-surfactant systems. The nonlinear sixth-order parabolic equation is expressed in a mixed form whereby a second-order (in space) parabolic equation and an algebraic fourth-order (in space) nonlinear equation are considered. The temporal discretization is chosen so that a discrete energy law can be established leading to unconditional energy stability. Additionally, we show that the numerical method is unconditionally uniquely solvable. We conclude with several numerical experiments demonstrating the unconditional stability and first-order accuracy of the proposed method.

Amanda E. Diegel & Natasha S. Sharma. (2023). Unconditional Energy Stability and Solvability for a C0 Interior Penalty Method for a Sixth-Order Equation Modeling Microemulsions . International Journal of Numerical Analysis and Modeling. 20 (4). 459-477. doi:10.4208/ijnam2023-1019
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