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Keywords
(19)
Boundary Value Problem
Finite Element
Finite Element Method
Least Square
Method of Characteristics
Mixed Finite Element
Mixed Finite Element Method
Saddle Point
Shallow Water
Singular Perturbation Problem
Stokes Equation
Stokes Flow
time discretization
Type System
Vanishing Viscosity
Velocity Field
Water Level
First Order
First Order System Least Squares
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(1)
Stabilized CrouzeixRaviart element for the DarcyStokes problem
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FirstOrder System LeastSquares for DarcyStokes Flow
FirstOrder System LeastSquares for DarcyStokes Flow,10.1137/050638163,Siam Journal on Numerical Analysis,Garvin Danisch,Gerhard Starke
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FirstOrder System LeastSquares for DarcyStokes Flow
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Citations: 3
)
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Garvin Danisch
,
Gerhard Starke
The subject of this paper is a firstorder system leastsquares formulation for the
Stokes equation
which remains uniformly valid in the limit of vanishing viscosity. For this socalled DarcyStokes flow problem we establish continuity and coercivity of the corresponding leastsquares functional in appropriate norms. Two types of
finite element
spaces for the approximation of the
velocity field
are investigated in detail: the wellknown RaviartThomas elements and an element recently introduced by Mardal, Tai and Winther specifically for mixed approaches to DarcyStokes flow. The computational results derived with nexttolowest order RaviartThomas elements as well as the MardalTaiWinther elements confirm the analysis. 1. Introduction. Our purpose in this paper is to present a leastsquares
finite element method
for DarcyStokes flow which remains valid for arbitrarily small vis cosity. This type of
singular perturbation problem
was studied before in (11) where a successful
mixed finite element
approach is presented. The mixed variational formula tion of (11) is of
saddle point
structure with its wellknown limitation on the admissible combinations of
finite element
spaces. One of the motivations for the development of the leastsquares approach presented in this paper is the greater flexibility in the choice of
finite element
spaces which is not restricted by a compatibility condition. In the limit of vanishing viscosity, our leastsquares formulation turns into the one proposed in (8). The approach in (8) constructs approximations for the pressure and the velocity in H1() and H(div,), respectively. In the viscous case, however, an approximation for the velocity is sought in H1() 2 instead. This is achieved by an augmentation with a leastsquares functional along the edges of the triangulation over the jump of the tangential component and by introducing the velocity gradient as an additional variable. The case of small viscosity is handled by an appropriate weighting of the components in the leastsquares functional. Our main motivation for this work comes from the treatment of
shallow water
systems treated with the
method of characteristics
for time discretization. In this context, linearization of the boundary value problems at each timestep leads to flow problems of DarcyStokes type.
Shallow water
flow is described by the scalar
water level
and by the velocity field. These process variables are directly approximated by the firstorder system leastsquares formulation treated in this paper. The exten sion to
shallow water
systems including a viscosity term is therefore straightforward. For
vanishing viscosity
that approach reduces to the firstorder system leastsquares method investigated in (14). Among the most popular methods for the case µ = 0 is the RaviartThomas
mixed finite element method
which couples, for example, lowestorder RaviartThomas elements for the flux with piecewise constant functions for the scalar variable. This approach is well studied in the case of the linear firstorder Darcytype system (see e.g. (4, Section III.5)) as well as for the
shallow water
system without viscosity (see
Journal:
Siam Journal on Numerical Analysis  SIAM J NUMER ANAL
, vol. 45, no. 2, pp. 731745, 2007
DOI:
10.1137/050638163
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Citation Context
(1)
...Another nonconforming approach is [9], while a least squares formulation based on the nonconforming element introduced in [20], was presented in [
11
]...
Gerard Awanou
.
Robustness of a Spline Element Method with Constraints
References
(13)
Finite Element Methods of LeastSquares Type
(
Citations: 127
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Pavel B. Bochev
,
Max D. Gunzburger
Journal:
Siam Review  SIAM REV
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Computing and Visualization in Science
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FirstOrder System Least Squares for SecondOrder Partial Differential Equations: Part I
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Siam Journal on Numerical Analysis  SIAM J NUMER ANAL
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Mesh adaptation strategies for shallow water flow
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M. Marrocu
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Journal:
International Journal for Numerical Methods in Fluids  INT J NUMER METHOD FLUID
, vol. 31, no. 2, pp. 497512, 1999
Analysis of VelocityFlux LeastSquares Principles for the NavierStokes Equations: Part II
(
Citations: 28
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Pavel Bochev
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Thomas A. Manteuffel
,
Stephen F. McCormick
Journal:
Siam Journal on Numerical Analysis  SIAM J NUMER ANAL
, vol. 36, no. 4, 1999
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Citations
(3)
Superconvergence analysis of FEMs for the StokesDarcy system
(
Citations: 2
)
Wenbin Chen
,
Puying Chen
,
Max Gunzburger
,
Ningning Yan
Journal:
Mathematical Methods in The Applied Sciences  MATH METH APPL SCI
, 2010
Superconvergence analysis of FEMs for the StokesDarcy system
(
Citations: 2
)
Wenbin Chen
Published in 2009.
Robustness of a Spline Element Method with Constraints
(
Citations: 2
)
Gerard Awanou
Journal:
Journal of Scientific Computing
, vol. 36, no. 3, pp. 421432, 2008