Error analysis of a mixed finite element method for a Cahn–Hilliard–Hele–Shaw system
We present and analyze a mixed finite element numerical scheme for the Cahn–Hilliard–Hele–Shaw equation, a modified Cahn–Hilliard equation coupled with the Darcy flow law. This numerical scheme was first reported in Feng and Wise (SIAM J Numer Anal 50:1320–1343, 2012 ), with the weak convergence to...
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| Published in | Numerische Mathematik Vol. 135; no. 3; pp. 679 - 709 |
|---|---|
| Main Authors | , , , |
| Format | Journal Article |
| Language | English |
| Published |
Berlin/Heidelberg
Springer Berlin Heidelberg
01.03.2017
Springer Nature B.V |
| Subjects | |
| Online Access | Get full text |
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| Abstract | We present and analyze a mixed finite element numerical scheme for the Cahn–Hilliard–Hele–Shaw equation, a modified Cahn–Hilliard equation coupled with the Darcy flow law. This numerical scheme was first reported in Feng and Wise (SIAM J Numer Anal 50:1320–1343,
2012
), with the weak convergence to a weak solution proven. In this article, we provide an optimal rate error analysis. A convex splitting approach is taken in the temporal discretization, which in turn leads to the unique solvability and unconditional energy stability. Instead of the more standard
ℓ
∞
(
0
,
T
;
L
2
)
∩
ℓ
2
(
0
,
T
;
H
2
)
error estimate, we perform a discrete
ℓ
∞
(
0
,
T
;
H
1
)
∩
ℓ
2
(
0
,
T
;
H
3
)
error estimate for the phase variable, through an
L
2
inner product with the numerical error function associated with the chemical potential. As a result, an unconditional convergence (for the time step
τ
in terms of the spatial resolution
h
) is derived. The nonlinear analysis is accomplished with the help of a discrete Gagliardo–Nirenberg type inequality in the finite element space, gotten by introducing a discrete Laplacian
Δ
h
of the numerical solution, such that
Δ
h
ϕ
∈
S
h
, for every
ϕ
∈
S
h
, where
S
h
is the finite element space. |
|---|---|
| AbstractList | We present and analyze a mixed finite element numerical scheme for the Cahn–Hilliard–Hele–Shaw equation, a modified Cahn–Hilliard equation coupled with the Darcy flow law. This numerical scheme was first reported in Feng and Wise (SIAM J Numer Anal 50:1320–1343, 2012), with the weak convergence to a weak solution proven. In this article, we provide an optimal rate error analysis. A convex splitting approach is taken in the temporal discretization, which in turn leads to the unique solvability and unconditional energy stability. Instead of the more standard ℓ ∞ ( 0 , T ; L 2 ) ∩ ℓ 2 ( 0 , T ; H 2 ) error estimate, we perform a discrete ℓ ∞ ( 0 , T ; H 1 ) ∩ ℓ 2 ( 0 , T ; H 3 ) error estimate for the phase variable, through an L 2 inner product with the numerical error function associated with the chemical potential. As a result, an unconditional convergence (for the time step τ in terms of the spatial resolution h) is derived. The nonlinear analysis is accomplished with the help of a discrete Gagliardo–Nirenberg type inequality in the finite element space, gotten by introducing a discrete Laplacian Δ h of the numerical solution, such that Δ h ϕ ∈ S h , for every ϕ ∈ S h , where S h is the finite element space. We present and analyze a mixed finite element numerical scheme for the Cahn–Hilliard–Hele–Shaw equation, a modified Cahn–Hilliard equation coupled with the Darcy flow law. This numerical scheme was first reported in Feng and Wise (SIAM J Numer Anal 50:1320–1343, 2012 ), with the weak convergence to a weak solution proven. In this article, we provide an optimal rate error analysis. A convex splitting approach is taken in the temporal discretization, which in turn leads to the unique solvability and unconditional energy stability. Instead of the more standard ℓ ∞ ( 0 , T ; L 2 ) ∩ ℓ 2 ( 0 , T ; H 2 ) error estimate, we perform a discrete ℓ ∞ ( 0 , T ; H 1 ) ∩ ℓ 2 ( 0 , T ; H 3 ) error estimate for the phase variable, through an L 2 inner product with the numerical error function associated with the chemical potential. As a result, an unconditional convergence (for the time step τ in terms of the spatial resolution h ) is derived. The nonlinear analysis is accomplished with the help of a discrete Gagliardo–Nirenberg type inequality in the finite element space, gotten by introducing a discrete Laplacian Δ h of the numerical solution, such that Δ h ϕ ∈ S h , for every ϕ ∈ S h , where S h is the finite element space. |
| Author | Wang, Cheng Liu, Yuan Chen, Wenbin Wise, Steven M. |
| Author_xml | – sequence: 1 givenname: Yuan surname: Liu fullname: Liu, Yuan organization: School of Mathematical Sciences, Fudan University – sequence: 2 givenname: Wenbin surname: Chen fullname: Chen, Wenbin organization: School of Mathematical Sciences, Fudan University – sequence: 3 givenname: Cheng surname: Wang fullname: Wang, Cheng organization: Mathematics Department, University of Massachusetts – sequence: 4 givenname: Steven M. surname: Wise fullname: Wise, Steven M. email: swise1@utk.edu organization: Mathematics Department, University of Tennessee |
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| Snippet | We present and analyze a mixed finite element numerical scheme for the Cahn–Hilliard–Hele–Shaw equation, a modified Cahn–Hilliard equation coupled with the... |
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| SubjectTerms | Chemical potential Convergence Error analysis Error functions Finite element method Mathematical analysis Mathematical and Computational Engineering Mathematical and Computational Physics Mathematical Methods in Physics Mathematics Mathematics and Statistics Nonlinear analysis Numerical Analysis Numerical and Computational Physics Simulation Spatial resolution Theoretical |
| Title | Error analysis of a mixed finite element method for a Cahn–Hilliard–Hele–Shaw system |
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