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Energy stable and convergent finite element schemes for the modified phase field crystal equation
- Source :
- ESAIM: Mathematical Modelling and Numerical Analysis. 50:1523-1560
- Publication Year :
- 2016
- Publisher :
- EDP Sciences, 2016.
-
Abstract
- We propose a space semi-discrete and a fully discrete finite element scheme for the modified phase field crystal equation (MPFC). The space discretization is based on a splitting method and consists in a Galerkin approximation which allows low order (piecewise linear) finite elements. The time discretization is a second-order scheme which has been introduced by Gomez and Hughes for the Cahn-Hilliard equation. The fully discrete scheme is shown to be unconditionally energy stable and uniquely solvable for small time steps, with a smallness condition independent of the space step. Using energy estimates, we prove that in both cases, the discrete solution converges to the unique energy solution of the MPFC equation as the discretization parameters tend to 0. Using a Lojasiewicz inequality, we also establish that the discrete solution tends to a stationary solution as time goes to infinity. Numerical simulations illustrate the theoretical results.
- Subjects :
- Discretization
65P40
Finite elements
010103 numerical & computational mathematics
Space (mathematics)
01 natural sciences
Piecewise linear function
second-order schemes
[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP]
Order (group theory)
0101 mathematics
Galerkin method
Mathematics
Numerical Analysis
74N05
Lojasiewicz inequality MSC 2010: 65M60
Applied Mathematics
Mathematical analysis
Finite element method
010101 applied mathematics
Computational Mathematics
gradient-like systems
Modeling and Simulation
Scheme (mathematics)
[MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Analysis
Energy (signal processing)
82C26
Subjects
Details
- ISSN :
- 12903841 and 0764583X
- Volume :
- 50
- Database :
- OpenAIRE
- Journal :
- ESAIM: Mathematical Modelling and Numerical Analysis
- Accession number :
- edsair.doi.dedup.....4ab3066379cd3c077fc43b05bfa1afd8