A DDFV method for a Cahn−Hilliard/Stokes phase field model with dynamic boundary conditions

Boyer, Franck; Nabet, Flore

doi:10.1051/m2an/2016073

Cited by 10 publications

(9 citation statements)

References 41 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Besides, in the situation of conforming triangle meshes and nonconforming Cartesian meshes, we can select the stabilization parameter λ = 0, because in this case the discrete inf‐sup condition holds. Finally, we conclude this section with some useful inequalities . Lemma The mean ‐ value projection

ℙ_{boldm}^{scriptT}

, the center‐value projection

ℙ_{boldc}^{scriptT}

and the mean ‐ value projection

ℙ_{boldm}^{frakturD}

satisfy the following inequalities ,

| {(‖‖|, \nabla^{D} ℙ_{m}^{T} boldv)}_{2} \leq C | {(‖‖|, \nabla boldv)}_{2}, \forall v \in {((), H^{1} ((), Ω))}^{2},

| {(‖‖|, \nabla^{D} ℙ_{c}^{T} boldv - \nabla boldv)}_{2} \leq {italicCh}_{scriptT} | {(‖‖|, \nabla boldv)}_{H^{1}}, \forall v \in {((), H^{2} ((), Ω))}^{2},

{(‖‖, ℙ_{c}^{T} boldv - boldv)}_{2} \leq {italicCh}_{scriptT} | {(‖‖|, \nabla boldv)}_{H^{1}}, \forall v \in {((), H^{2} ((), Ω))}^{2},

{(‖‖, ℙ_{m}^{D} q - q)}_{2} \leq {italicCh}_{scriptT} {(‖‖, \nabla q)}_{2}, \forall q \in

…”

Section: The Ddfv Frameworkmentioning

confidence: 96%

“…In this section, we state some basic notations briefly for the DDFV method (refer to for a detailed description). The DDFV method contains three meshes: the primal mesh

\overset{true‾}{frakturM}

, the dual mesh

\overset{true‾}{M^{*}}

, and the diamond mesh

D

.…”

Section: The Ddfv Frameworkmentioning

confidence: 99%

“…From the definition of the discrete gradient operator

\nabla^{frakturD}

, we know that

(E_{u}, {(\cdot \cdot)}_{\nabla^{frakturD}})

is an inner product space, and for any

{boldu}^{scriptT} \in {double-struckE}_{boldu}

, we denote the corresponding norm by

| {(‖‖|, \nabla^{D} u^{T})}_{2}

. On the other hand, by the following discrete Korn inequality in .Lemma (Discrete Korn inequality). For any

{boldu}^{scriptT} \in {double-struckE}_{boldu}

, we have

| {(‖‖|, D^{D} u^{T})}_{2} \leq | {(‖‖|, \nabla^{D} u^{T})}_{2} \leq \sqrt{2} | {(‖‖|, D^{D} u^{T})}_{2} .

We know that

(E_{u}, {(\cdot \cdot)}_{{normalD}^{frakturD}})

is also an inner product space, and we denote the corresponding norm by

| {(‖‖|, D^{D} u^{T})}_{2}

for any

{boldu}^{scriptT} \in {double-struckE}_{boldu}

.Lemma (Discrete Stokes formula).…”

Section: The Ddfv Frameworkmentioning

confidence: 99%

See 2 more Smart Citations

The discrete duality finite volume method for a class of quasi‐Newtonian Stokes flows

Chen

et al. 2019

Numerical Methods Partial

View full text Add to dashboard Cite

In this paper, we propose a discrete duality finite volume (DDFV) scheme for the incompressible quasi-Newtonian Stokes equation. The DDFV method is based on the use of discrete differential operators which satisfy some duality properties analogous to their continuous counterparts in a discrete sense. The DDFV method has a great ability to handle general geometries and meshes. In addition, every component of the velocity gradient can be reconstructed directly, which makes it suitable to deal with the nonlinear terms in the quasi-Newtonian Stokes equation. We prove that the proposed DDFV scheme is uniquely solvable and of first-order convergence in the discrete L 2 -norms for the velocity, the strain rate tensor, and the pressure, respectively. Ample numerical tests are provided to highlight the performance of the proposed DDFV scheme and to validate the theoretical error analysis, in particular on locally refined nonconforming and polygonal meshes. KEYWORDS discrete duality finite volume method, distorted, nonconforming and polygonal meshes, quasi-Newtonian Stokes equation 1

show abstract

ℙ_{boldm}^{scriptT}

, the center‐value projection

ℙ_{boldc}^{scriptT}

and the mean ‐ value projection

ℙ_{boldm}^{frakturD}

satisfy the following inequalities ,

| {(‖‖|, \nabla^{D} ℙ_{m}^{T} boldv)}_{2} \leq C | {(‖‖|, \nabla boldv)}_{2}, \forall v \in {((), H^{1} ((), Ω))}^{2},

| {(‖‖|, \nabla^{D} ℙ_{c}^{T} boldv - \nabla boldv)}_{2} \leq {italicCh}_{scriptT} | {(‖‖|, \nabla boldv)}_{H^{1}}, \forall v \in {((), H^{2} ((), Ω))}^{2},

{(‖‖, ℙ_{c}^{T} boldv - boldv)}_{2} \leq {italicCh}_{scriptT} | {(‖‖|, \nabla boldv)}_{H^{1}}, \forall v \in {((), H^{2} ((), Ω))}^{2},

{(‖‖, ℙ_{m}^{D} q - q)}_{2} \leq {italicCh}_{scriptT} {(‖‖, \nabla q)}_{2}, \forall q \in

…”

Section: The Ddfv Frameworkmentioning

confidence: 96%

“…In this section, we state some basic notations briefly for the DDFV method (refer to for a detailed description). The DDFV method contains three meshes: the primal mesh

\overset{true‾}{frakturM}

, the dual mesh

\overset{true‾}{M^{*}}

, and the diamond mesh

D

.…”

Section: The Ddfv Frameworkmentioning

confidence: 99%

“…From the definition of the discrete gradient operator

\nabla^{frakturD}

, we know that

(E_{u}, {(\cdot \cdot)}_{\nabla^{frakturD}})

is an inner product space, and for any

{boldu}^{scriptT} \in {double-struckE}_{boldu}

, we denote the corresponding norm by

| {(‖‖|, \nabla^{D} u^{T})}_{2}

. On the other hand, by the following discrete Korn inequality in .Lemma (Discrete Korn inequality). For any

{boldu}^{scriptT} \in {double-struckE}_{boldu}

, we have

| {(‖‖|, D^{D} u^{T})}_{2} \leq | {(‖‖|, \nabla^{D} u^{T})}_{2} \leq \sqrt{2} | {(‖‖|, D^{D} u^{T})}_{2} .

We know that

(E_{u}, {(\cdot \cdot)}_{{normalD}^{frakturD}})

is also an inner product space, and we denote the corresponding norm by

| {(‖‖|, D^{D} u^{T})}_{2}

for any

{boldu}^{scriptT} \in {double-struckE}_{boldu}

.Lemma (Discrete Stokes formula).…”

Section: The Ddfv Frameworkmentioning

confidence: 99%

See 1 more Smart Citation

The discrete duality finite volume method for a class of quasi‐Newtonian Stokes flows

Chen

et al. 2019

Numerical Methods Partial

View full text Add to dashboard Cite

show abstract

“…In [5] we give a more complex method called DDFV method to solve this problem which enables us to use more general meshes without orthogonality condition as for example non-conforming mesh. We also propose an original DDFV scheme to study the Cahn-Hilliard/Stokes phase field model.…”

Section: Preferential Attraction By the Wallmentioning

confidence: 99%

Convergence of a finite-volume scheme for the Cahn–Hilliard equation with dynamic boundary conditions

Nabet¹

2015

IMA J Numer Anal

Self Cite

View full text Add to dashboard Cite

show abstract

“…Remark 4.5) was analyzed in [37], along with numerical computations of singular solutions in one space dimension. The numerical analysis and numerical computation of related problems involving dynamic boundary conditions and regular nonlinearities were also considered in, e.g., [1,6,19,20,26,30,31,36]. Cahn-Hilliard type equations with logarithmic potential and classical boundary conditions have also drawn a lot of interest, cf., e.g.…”

mentioning

confidence: 99%

A doubly splitting scheme for the Caginalp system with singular potentials and dynamic boundary conditions

Langa¹,

Pierre²

2021

Discrete &Amp; Continuous Dynamical Systems - S

View full text Add to dashboard Cite

We propose a time semi-discrete scheme for the Caginalp phasefield system with singular potentials and dynamic boundary conditions. The scheme is based on a time splitting which decouples the equations and on a convex splitting of the energy associated to the problem. The scheme is unconditionally uniquely solvable and the energy is nonincreasing if the time step is small enough. The discrete solution is shown to converge to the energy solution of the problem as the time step tends to 0. The proof involves a multivalued operator and a monotonicity argument. This approach allows us to compute numerically singular solutions to the problem.

show abstract

A DDFV method for a Cahn−Hilliard/Stokes phase field model with dynamic boundary conditions

Cited by 10 publications

References 41 publications

The discrete duality finite volume method for a class of quasi‐Newtonian Stokes flows

The discrete duality finite volume method for a class of quasi‐Newtonian Stokes flows

Convergence of a finite-volume scheme for the Cahn–Hilliard equation with dynamic boundary conditions

A doubly splitting scheme for the Caginalp system with singular potentials and dynamic boundary conditions

Contact Info

Product

Resources

About