Error estimates for a finite element–finite volume discretization of convection–diffusion equations

Deuring, Paul; Mildner, Marcus

doi:10.1016/j.apnum.2011.01.007

Cited by 4 publications

(9 citation statements)

References 31 publications

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“…Thus our scheme does allow to maintain accuracy. Reference [12] generalizes earlier results from [13], where the case b constant, m = 0, Γ D = ∂Ω, f D = 0, a = −ν (δ jk ) 1≤j,k≤2 is considered.…”

Section: Introductionmentioning

confidence: 89%

L²-stability of a finite element – finite volume discretization of convection-diffusion-reaction equations with nonhomogeneous mixed boundary conditions

Deuring

Eymard

2017

ESAIM: M2AN

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Abstract.We consider a time-dependent and a steady linear convection-diffusion-reaction equation whose coefficients are nonconstant. Boundary conditions are mixed (Dirichlet and Robin−Neumann) and nonhomogeneous. Both the unsteady and the steady problem are approximately solved by a combined finite element -finite volume method: the diffusion term is discretized by Crouzeix−Raviart piecewise linear finite elements on a triangular grid, and the convection term by upwind barycentric finite volumes. In the unsteady case, the implicit Euler method is used as time discretization. This scheme is shown to be unconditionally L 2 -stable, uniformly with respect to diffusion, except if the Robin−Neumann boundary condition is inhomogeneous and the convective velocity is tangential at some points of the Robin−Neumann boundary. In that case, a negative power of the diffusion coefficient arises. As is shown by a counterexample, this exception cannot be avoided.Mathematics Subject Classification. 65M12, 65M60.

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Section: Introductionmentioning

confidence: 89%

L²-stability of a finite element – finite volume discretization of convection-diffusion-reaction equations with nonhomogeneous mixed boundary conditions

Deuring

Eymard

2017

ESAIM: M2AN

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“…[2, section 7], [3, chapter 4.4] tested this FE–FV method in the case of high‐speed compressible Navier–Stokes flows in complex geometries and obtained very satisfactory results. In , we applied this FE–FV method to problem (1.1)–(1.3), using the implicit Euler method as time discretization. Under the assumptions

Γ_{D} = \partial Ω, f_{D} = 0

(homogeneous Dirichlet boundary conditions), m = 0 (no reaction term),

a (x, t) = ν {(δ_{j k})}_{1 \leq j, k \leq 2}

(Laplace operator with diffusion coefficient

ν \in (0, \infty)

b \in ℝ^{2} \ {0}

with

| b | = 1

(constant convective velocity), we derived a bound for the

L^{\infty} (L^{2})

‐ and the

L^{2} (H^{1})

‐error arising with this method.…”

Section: Introductionmentioning

confidence: 99%

“…In the work at hand, we are going to show that the numerical part of our theory in remains valid if the convective velocity b depends on the space and the time variable, if the highest‐order term

- ν Δ u

is replaced by the second order elliptic operator

- div (a (x, t) \nabla_{x} u (x, t))

, with the symmetric positive definite matrix a also depending on x and t , and if the homogeneous Dirichlet boundary conditions in are replaced by inhomogeneous ones on Γ D and by inhomogeneous Robin–Neumann conditions on Γ N .…”

Section: Introductionmentioning

confidence: 99%

“…These functions are just required to possess enough smoothness so that the discrete problem (2.10), (2.11) makes sense. In , we addressed the question as to which assumptions on the data are needed to obtain a solution of (1.1)–(1.3) as specified above, but as we may recall, this reference only deals with the special case

Γ_{D} = \partial Ω, Γ_{N} = \emptyset, f_{D} = 0

(homogeneous Dirichlet boundary conditions),

b \in ℝ^{2} \ {0}

and suitable initial data

u^{(0)}

. In such a situation, the theory in implies that the solution w of (1.1)–(1.3) does in fact possess the regularity properties required above.…”

Section: Introductionmentioning

confidence: 99%

“…We remark that the role played by the parameter ν in is now taken by the quantity

ν - α_{0} C_{1}^{- 2}

, where C 1 is the constant from a Poincaré's inequality (Theorem 2.1). This is because in (1.7) and (1.9), we require

{div}_{x} b / 2 + m \geq - α_{0}

and

div B / 2 + M \geq - α_{0}

, respectively, instead of only

{div}_{x} b / 2 + m \geq 0

and

div B / 2 + M \geq 0

.…”

Section: Introductionmentioning

confidence: 99%

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A finite element–finite volume discretization of convection‐diffusion‐reaction equations with nonhomogeneous mixedboundary conditions: Error estimates

Deuring

2016

Numerical Methods Partial

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We consider a time‐dependent and a steady linear convection‐diffusion‐reaction equation whose coefficients are nonconstant. Boundary conditions are mixed (Dirichlet and Robin–Neumann) and nonhomogeneous. Both the unsteady and the steady problem are approximately solved by a combined finite element–finite volume method: the diffusion term is discretized by Crouzeix–Raviart piecewise linear finite elements on a triangular grid, and the convection term by upwind barycentric finite volumes. In the unsteady case, the implicit Euler method is used as time discretization. The L 2 ( H 1 ) ‐ and the L ∞ ( L 2 ) ‐error in the unsteady case and the H1‐error in the steady one are estimated against the data, in such a way that no parameter enters exponentially into the constants involved. © 2016Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1591–1621, 2016

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A Restrained Optimal Perturbation Method for Solving the Inverse Problem in Reverse Process of Convection Diffusion Equation

Wang

Zou

Zhao

2012

Advanced Intelligent Computing Theories and Applications. With Aspects of Artificial Intelligence

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Error estimates for a finite element–finite volume discretization of convection–diffusion equations

Cited by 4 publications

References 31 publications

L²-stability of a finite element – finite volume discretization of convection-diffusion-reaction equations with nonhomogeneous mixed boundary conditions

L²-stability of a finite element – finite volume discretization of convection-diffusion-reaction equations with nonhomogeneous mixed boundary conditions

A finite element–finite volume discretization of convection‐diffusion‐reaction equations with nonhomogeneous mixedboundary conditions: Error estimates

A Restrained Optimal Perturbation Method for Solving the Inverse Problem in Reverse Process of Convection Diffusion Equation

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Error estimates for a finite element–finite volume discretization of convection–diffusion equations

Cited by 4 publications

References 31 publications

L2-stability of a finite element – finite volume discretization of convection-diffusion-reaction equations with nonhomogeneous mixed boundary conditions

L2-stability of a finite element – finite volume discretization of convection-diffusion-reaction equations with nonhomogeneous mixed boundary conditions

A finite element–finite volume discretization of convection‐diffusion‐reaction equations with nonhomogeneous mixedboundary conditions: Error estimates

A Restrained Optimal Perturbation Method for Solving the Inverse Problem in Reverse Process of Convection Diffusion Equation

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L²-stability of a finite element – finite volume discretization of convection-diffusion-reaction equations with nonhomogeneous mixed boundary conditions

L²-stability of a finite element – finite volume discretization of convection-diffusion-reaction equations with nonhomogeneous mixed boundary conditions