Journal of Scientific Computing

, Volume 76, Issue 3, pp 1938–1967 | Cite as

A Uniquely Solvable, Energy Stable Numerical Scheme for the Functionalized Cahn–Hilliard Equation and Its Convergence Analysis

  • Wenqiang Feng
  • Zhen Guan
  • John Lowengrub
  • Cheng WangEmail author
  • Steven M. Wise
  • Ying Chen


We present and analyze a uniquely solvable and unconditionally energy stable numerical scheme for the Functionalized Cahn–Hilliard equation, including an analysis of convergence. One key difficulty associated with the energy stability is based on the fact that one nonlinear energy functional term in the expansion is neither convex nor concave. To overcome this subtle difficulty, we add two auxiliary terms to make the combined term convex, which in turns yields a convex–concave decomposition of the physical energy. As a result, both the unconditional unique solvability and the unconditional energy stability of the proposed numerical scheme are assured. In addition, a global in time \(H_{\mathrm{per}}^2\) stability of the numerical scheme is established at a theoretical level, which in turn ensures the full order convergence analysis of the scheme, which is the first such result in this field. To deal with an implicit 4-Laplacian term at each time step, we apply an efficient preconditioned steepest descent algorithm to solve the corresponding nonlinear systems in the finite difference set-up. A few numerical results are presented, which confirm the stability and accuracy of the proposed numerical scheme.


Functionalized Cahn–Hilliard equation Finite difference method Energy stability Convergence analysis Preconditioned steepest descent solver 

Mathematics Subject Classification

35K35 35K55 65M06 65M12 



JSL acknowledges partial support from NSF-CHE 1035218, NSF-DMR 1105409, NSF-DMS 1217273 and DMS-FRG 1507033. CW acknowledges partial support from NSF-DMS 1418689. SMW acknowledges partial support from NSF-DMS1418692 and NSF-DMS 1719854.


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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsThe University of TennesseeKnoxvilleUSA
  2. 2.Department of MathematicsThe University of CaliforniaIrvineUSA
  3. 3.Department of MathematicsThe University of MassachusettsNorth DartmouthUSA
  4. 4.Department of MathematicsDuke UniversityDurhamUSA

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