## Domain Decomposition Algorithms For Mixed Methods For Second Order Elliptic Problems (0)

Venue: | Math. Comp |

Citations: | 20 - 12 self |

### BibTeX

@ARTICLE{Chen_domaindecomposition,

author = {Zhangxin Chen and Richard E. Ewing and Raytcho Lazarov},

title = {Domain Decomposition Algorithms For Mixed Methods For Second Order Elliptic Problems},

journal = {Math. Comp},

year = {},

volume = {65},

pages = {467--490}

}

### OpenURL

### Abstract

. In this paper domain decomposition algorithms for mixed finite element methods for linear and quasilinear second order elliptic problems in IR 2 and IR 3 are developed. A convergence theory for two-level and multilevel Schwarz methods applied to the algorithms under consideration is given, and its extension to other substructuring methods such as vertex space and balancing domain decomposition methods is considered. It is shown that the condition number of these iterative methods is bounded uniformly from above in the same manner as in the theory of domain decomposition methods for conforming and nonconforming finite element methods for the same differential problems. Numerical experiments are presented to illustrate the present techniques. 1. Introduction. This is the second paper of a sequence where we develop and analyze efficient iterative algorithms for solving the linear system arising from mixed finite element methods for linear and quasilinear second order elliptic proble...

### Citations

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The finite element method for elliptic problems
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190 |
A mixed finite element method for 2nd order elliptic problems
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- 1975
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161 |
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Citation Context ...es and the number of decompositions. Hence, the experimental results coincide with the theory established before. An extension of the present approach to other substructuring methods such as those in =-=[5, 29, 38]-=- will be discussed in a forthcoming paper. Table 1. The condition number with ffi = H=4. 1=h 16 16 24 24 32 32 n 8 64 8 27 8 64 c(\Pi ) 5.86 5.12 6.08 6.67 6.51 6.81 # 8 9 9 8 8 9 Table 2. The conditi... |

109 |
Mixed Finite Elements
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- 1980
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108 |
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- 1985
(Show Context)
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77 |
Efficient rectangular mixed finite elements in two and three variables
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Citation Context ...nterior edges (d = 2) or faces (d = 3) e of E h . Let V h \Theta W h ae V \Theta W denote some standard mixed finite element space for second order elliptic problems defined over E h (see, e.g., [5], =-=[6]-=-, [7], [13], [33], [34], and [35]). This space is finite dimensional and defined locally on each element E 2 E h , so let V h (E) = V h j E and W h (E) = W h j E . The constraint V h ae V says that th... |

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Citation Context ...AROV divergence free vectors. This approach is limited to two space dimensions [22], [23], [24], [25], [26], [29], [30], [31], [39]. The other method is the so-called dual variable method [15], [16], =-=[17]-=-, [25], [26]. This approach makes use of a discretization of the flux operator (the coefficient times the gradient), which transfers the original saddle point problem to an elliptic problem for the sc... |

70 |
Mixed finite elements for second order elliptic problems in three variables
- Brezzi, Douglas, et al.
- 1987
(Show Context)
Citation Context ...all interior edges (d = 2) or faces (d = 3) e of E h . Let V h \Theta W h ae V \Theta W denote some standard mixed finite element space for second order elliptic problems defined over E h (see, e.g., =-=[5]-=-, [6], [7], [13], [33], [34], and [35]). This space is finite dimensional and defined locally on each element E 2 E h , so let V h (E) = V h j E and W h (E) = W h j E . The constraint V h ae V says th... |

46 | On the implementation of mixed methods as nonconforming methods for second order elliptic problems
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- 1995
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Citation Context ...e, the standard theory for the domain decomposition methods applied to nonconforming (even conforming) finite element methods applies to the mixed methods. Finally, bubble functions have been used in =-=[1, 2, 10]-=- to establish the equivalence between mixed finite element methods and certain nonconforming methods. The approach under consideration does not make use of bubble functions. The present approach is ex... |

43 |
Mixed finite elements in R3
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Citation Context ...t of all interior edges (d = 2) or faces (d = 3) e of Eh. Let Vh \Theta Wh ae V \Theta W denote some standard mixed finite element space for second-order elliptic problems defined over Eh (see, e.g., =-=[6, 7, 8, 14, 22, 34, 35, 36]-=-). This space is finite-dimensional and defined locally on each element E 2 Eh; so let Vh(E) = VhjE and Wh(E) = WhjE. The constraint Vh ae V says that the normal component of the members of Vh is cont... |

36 | Prismatic mixed finite elements for second order elliptic problems
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- 1989
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Citation Context ...ges (d = 2) or faces (d = 3) e of E h . Let V h \Theta W h ae V \Theta W denote some standard mixed finite element space for second order elliptic problems defined over E h (see, e.g., [5], [6], [7], =-=[13]-=-, [33], [34], and [35]). This space is finite dimensional and defined locally on each element E 2 E h , so let V h (E) = V h j E and W h (E) = W h j E . The constraint V h ae V says that the normal co... |

36 | Two-level additive Schwarz preconditioners for nonconforming finite element methods
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32 |
Analysis of mixed methods using conforming and nonconforming nite element methods
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- 1993
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32 |
Multilevel iterative methods for mixed finite element discretizations of elliptic problems
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- 1992
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Citation Context ...rtment of Energy under contract DE-ACOS-840R21400. 2 CHEN, EWING, AND LAZAROV divergence free vectors. This approach is limited to two space dimensions [22], [23], [24], [25], [26], [29], [30], [31], =-=[39]-=-. The other method is the so-called dual variable method [15], [16], [17], [25], [26]. This approach makes use of a discretization of the flux operator (the coefficient times the gradient), which tran... |

31 |
Analysis and test of a local domain decomposition preconditioner
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- 1991
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Citation Context ...HEN, EWING, AND LAZAROV The simplest choice for D i is the diagonal matrix with diagonal elements equal to the reciprocal of the number of substructures with which the degree of freedom is associated =-=[19], [27]. We-=- now define a "coarse space" N h (\Gamma H ) ae N h (\Gamma) by N h (\Gamma H ) = fv 2 N h (\Gamma) : v = n X i=1 L i D i '; ' 2 Range Y i g: As in [27], v 2 N h (\Gamma) is said balanced if... |

29 |
alternating and iterative refinement methods for mixed formulations of elliptic problems, part I: Algorithms and numerical results, tech
- Schwarz
- 1991
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Citation Context ... for the flux variable (the gradient of the scalar unknown times the coefficient of the differential problems) on the space of divergencefree vectors. This approach is limited to two space dimensions =-=[24, 25, 26, 27, 28, 31, 32, 40]-=-. The other method is the so-called dual variable method [16, 18, 19, 27, 28], This approach makes use of a discretization of the flux operator (the coefficient times the gradient), which transfers th... |

28 |
An optimal domain decomposition preconditioner for the nite element solution of linear elasticity problems
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- 1992
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Citation Context ...en as follows: Find / h 2 N h (\Gamma) such that (9.2) b(/ h ; ' h ) = f B (' h ); 8' h 2 N h (\Gamma); where the linear functional f B corresponds to FB . 9.1. Smith's algorithm. The Smith algorithm =-=[37]-=- is an additive Schwarz algorithm applied to the interface problem (9.2). Note that \Gamma has a natural decomposition from f\Omega i g, i.e., \Gamma = [ n i=1 \Gamma i where \Gamma i = @\Omega i n @\... |

25 |
Analysis of the Schwarz algorithm for mixed finite element methods
- EWING, WANG
- 1992
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Citation Context ... coefficient. Partly supported by the Department of Energy under contract DE-ACOS-840R21400. 2 CHEN, EWING, AND LAZAROV divergence free vectors. This approach is limited to two space dimensions [22], =-=[23]-=-, [24], [25], [26], [29], [30], [31], [39]. The other method is the so-called dual variable method [15], [16], [17], [25], [26]. This approach makes use of a discretization of the flux operator (the c... |

24 |
Parallel domain decomposition method for mixed nite elements for elliptic partial di erential equations
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- 1991
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Citation Context ...ING, AND LAZAROV divergence free vectors. This approach is limited to two space dimensions [22], [23], [24], [25], [26], [29], [30], [31], [39]. The other method is the so-called dual variable method =-=[15]-=-, [16], [17], [25], [26]. This approach makes use of a discretization of the flux operator (the coefficient times the gradient), which transfers the original saddle point problem to an elliptic proble... |

24 |
Domain decomposition and iterative refinement methods for mixed finite element discretizations of elliptic problems
- Mathew
- 1989
(Show Context)
Citation Context ...ported by the Department of Energy under contract DE-ACOS-840R21400. 2 CHEN, EWING, AND LAZAROV divergence free vectors. This approach is limited to two space dimensions [22], [23], [24], [25], [26], =-=[29]-=-, [30], [31], [39]. The other method is the so-called dual variable method [15], [16], [17], [25], [26]. This approach makes use of a discretization of the flux operator (the coefficient times the gra... |

23 |
A parallel iterative procedure applicable to the approximate solution of second order partial di erential equations by mixed nite element methods
- Douglas, Leme, et al.
- 1993
(Show Context)
Citation Context ...ch is established on the domain decomposition methods for a positive definite problem for the scalar and Lagrange multiplier. Recently, an iterative procedure based on domain decomposition techniques =-=[20]-=- was proposed for solving the linear system for the scalar, the flux, and the Lagrange multiplier, but the convergence analysis is restricted to use of subdomains as small as individual finite element... |

23 | Two-level Schwarz methods for nonconforming nite elements and discontinuous coe cients
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- 1993
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Citation Context ...1 and N2 are the number of the adjacent midpoints (x0 j� y0 j) and (x00 j � y00 j ) to (xi�yi) of the edges in @Ek;1 and the edges on @ of the elements in Ek;1, respectively. Alternatively, following =-=[37]-=-, Ik k;1 : Nk;1 ! Nk can be equivalently de ned by (3.16a) (3.16b) (3.16c) Ik k;1v(xi� yi)=v(xi� yi)� Ik k;1v(xi�yi) = 1 I k k;1v(xi�yi) = 1 X N1 (xi�yi)2Kj N2 X j i =1� 2� 3� vjKj (xi�yi) if (xi�yi) ... |

23 | Equivalence between and multigrid algorithms for mixed and nonconforming methods for second order elliptic problems, IMA
- Chen
- 1994
(Show Context)
Citation Context ...lyze efficient iterative algorithms for solving the linear system arising from mixed finite element methods for linear and quasilinear second-order elliptic problems in R2 and R3 . In the first paper =-=[12]-=-, a new approach for developing multigrid algorithms for the mixed finite element methods was introduced. It was first shown that the mixed finite element formulation can be algebraically condensed to... |

18 |
Acceleration of domain decomposition algorithms for mixed nite elements by multilevel methods
- Glowinski, Kinton, et al.
- 1990
(Show Context)
Citation Context .... Partly supported by the Department of Energy under contract DE-ACOS-840R21400. 2 CHEN, EWING, AND LAZAROV divergence free vectors. This approach is limited to two space dimensions [22], [23], [24], =-=[25]-=-, [26], [29], [30], [31], [39]. The other method is the so-called dual variable method [15], [16], [17], [25], [26]. This approach makes use of a discretization of the flux operator (the coefficient t... |

17 |
Domain decomposition methods for nonconforming nite elements spaces of Lagrange type
- Cowsar
- 1993
(Show Context)
Citation Context ...e estimate the two constants C and ae(). For this, let R h be the nodal interpolation operator into N h , and let UH be the conforming space of linear polynomials associated with EH . Then, following =-=[18]-=-, we define N 0 h as follows: (3.14) N 0 h = fv 2 N h : v = R h '; ' 2 UH g: To give the second example, let E h be the finest triangulation and let E h = EHJ for some Js1 where EHk = E k (H k = 2 \Ga... |

16 |
Analysis of multilevel decomposition iterative methods for mixed finite element methods, RAIROModél
- Ewing, Wang
- 1994
(Show Context)
Citation Context ...icient. Partly supported by the Department of Energy under contract DE-ACOS-840R21400. 2 CHEN, EWING, AND LAZAROV divergence free vectors. This approach is limited to two space dimensions [22], [23], =-=[24]-=-, [25], [26], [29], [30], [31], [39]. The other method is the so-called dual variable method [15], [16], [17], [25], [26]. This approach makes use of a discretization of the flux operator (the coeffic... |

15 | Local refinement via domain decomposition techniques for mixed finite element methods with regular Raviart-Thomas elements - Ewing, Lazarov, et al. - 1990 |

14 |
Mixed finite element methods for quasilinear second order elliptic problems
- Milner
- 1985
(Show Context)
Citation Context ...) @En@\Omega = (g; v \Delta ) \Gamma 1 ; 8v 2 ~ V h ; (8.2b) X E2Eh (oe h \DeltasE ; ��) @En@\Omega = 0; 8�� 2 L h ; (8.2c) where ff h (u h ) = P h ff(u h ). It has a unique solution [8], [10]=-=, [12], [32]-=-. A linearized version of (8.2) can be constructed as follows. Starting from any (oe 0 h ; u 0 h ;s0 h ) 2 ~ V h \Theta W h \Theta L h , we DOMAIN DECOMPOSITION ALGORITHMS FOR MIXED METHODS 25 constru... |

12 | Balancing domain decomposition: theory and performance in two and three dimensions
- Mandel, Brezina
(Show Context)
Citation Context ...the block unknowns ` j and �� j are void and the jth block equation is taken out of (9.7) and (9.8). The convergence property of this balancing algorithm can be obtained from the corresponding res=-=ult [28]-=- for the conforming elements using the result in Lemma 18. DOMAIN DECOMPOSITION ALGORITHMS FOR MIXED METHODS 29 Theorem 19. There is a constant C independent of h and H such that the condition number ... |

12 |
An application of the abstract multilevel theory to nonconforming finite element methods
- Vassilevski, Wang
- 1995
(Show Context)
Citation Context ...\Delta \Delta ; J; where we assume that I J = I J is the identity operator on N h , and C 1 satisfies (3.28) 0 ! s ks(C 1 H 2 k ) \Gamma1 ; where s k is the largest eigenvalue of S 0 k . It was shown =-=[38]-=- that there is a constant C 1 independent of k such that this inequality is indeed satisfied. So the operator S k is well defined. We are now ready to define the multilevel algorithms for (3.5) and th... |

12 |
Mixed and nonconforming nite element methods: implementation, postprocessing and error
- Arnold, Brezzi
- 1985
(Show Context)
Citation Context ...nce, the standard theory for the domain decomposition methods applied to nonconforming (even conforming) nite element methods applies to the mixed methods. Finally, bubble functions have been used in =-=[1, 2, 10]-=- to establish the equivalence between mixed nite element methods and certain nonconforming methods. The approach under consideration does not make use of bubble functions. The present approach is expl... |

12 |
Two families of mixed nite elements for second order elliptic problems
- Brezzi, Douglas, et al.
- 1985
(Show Context)
Citation Context ...e� let @Eh denote the set of all interior edges (d =2)or faces (d =3)e of Eh. Let Vh Wh V W denote some standard mixed nite element space for second-order elliptic problems de ned over Eh (see, e.g., =-=[6, 7, 8, 14, 22, 34, 35, 36]-=-). This space is nite-dimensional and de ned locally on each element E 2Eh� so let Vh(E) = VhjE and Wh(E) = WhjE. The constraint Vh V says that the normal component of the members of Vh is continuous ... |

11 |
Nedelec, A new family of mixed finite elements
- C
(Show Context)
Citation Context ...or faces (d = 3) e of E h . Let V h \Theta W h ae V \Theta W denote some standard mixed finite element space for second order elliptic problems defined over E h (see, e.g., [5], [6], [7], [13], [33], =-=[34]-=-, and [35]). This space is finite dimensional and defined locally on each element E 2 E h , so let V h (E) = V h j E and W h (E) = W h j E . The constraint V h ae V says that the normal component of t... |

10 | Approximation of coefficients in hybrid and mixed methods for nonlinear parabolic problems
- Chen, Douglas
- 1991
(Show Context)
Citation Context ...ltasE ) @En@\Omega = (g; v \Delta ) \Gamma 1 ; 8v 2 ~ V h ; (8.2b) X E2Eh (oe h \DeltasE ; ��) @En@\Omega = 0; 8�� 2 L h ; (8.2c) where ff h (u h ) = P h ff(u h ). It has a unique solution [8]=-=, [10], [12]-=-, [32]. A linearized version of (8.2) can be constructed as follows. Starting from any (oe 0 h ; u 0 h ;s0 h ) 2 ~ V h \Theta W h \Theta L h , we DOMAIN DECOMPOSITION ALGORITHMS FOR MIXED METHODS 25 c... |

10 |
E cient rectangular mixed nite elements in two and three space variables
- Brezzi, Douglas, et al.
- 1987
(Show Context)
Citation Context ...e� let @Eh denote the set of all interior edges (d =2)or faces (d =3)e of Eh. Let Vh Wh V W denote some standard mixed nite element space for second-order elliptic problems de ned over Eh (see, e.g., =-=[6, 7, 8, 14, 22, 34, 35, 36]-=-). This space is nite-dimensional and de ned locally on each element E 2Eh� so let Vh(E) = VhjE and Wh(E) = WhjE. The constraint Vh V says that the normal component of the members of Vh is continuous ... |

9 |
BDM mixed methods for a nonlinear elliptic problem
- Chen
- 1994
(Show Context)
Citation Context ... v \DeltasE ) @En@\Omega = (g; v \Delta ) \Gamma 1 ; 8v 2 ~ V h ; (8.2b) X E2Eh (oe h \DeltasE ; ��) @En@\Omega = 0; 8�� 2 L h ; (8.2c) where ff h (u h ) = P h ff(u h ). It has a unique soluti=-=on [8], [10]-=-, [12], [32]. A linearized version of (8.2) can be constructed as follows. Starting from any (oe 0 h ; u 0 h ;s0 h ) 2 ~ V h \Theta W h \Theta L h , we DOMAIN DECOMPOSITION ALGORITHMS FOR MIXED METHOD... |

9 |
Mixed nite elements for second order elliptic problems in three variables
- Brezzi, Douglas, et al.
- 1987
(Show Context)
Citation Context ...e� let @Eh denote the set of all interior edges (d =2)or faces (d =3)e of Eh. Let Vh Wh V W denote some standard mixed nite element space for second-order elliptic problems de ned over Eh (see, e.g., =-=[6, 7, 8, 14, 22, 34, 35, 36]-=-). This space is nite-dimensional and de ned locally on each element E 2Eh� so let Vh(E) = VhjE and Wh(E) = WhjE. The constraint Vh V says that the normal component of the members of Vh is continuous ... |

8 |
A new family of mixed finite element spaces over rectangles
- Douglas, Wang
- 1993
(Show Context)
Citation Context ...t of all interior edges (d = 2) or faces (d = 3) e of Eh. Let Vh \Theta Wh ae V \Theta W denote some standard mixed finite element space for second-order elliptic problems defined over Eh (see, e.g., =-=[6, 7, 8, 14, 22, 34, 35, 36]-=-). This space is finite-dimensional and defined locally on each element E 2 Eh; so let Vh(E) = VhjE and Wh(E) = WhjE. The constraint Vh ae V says that the normal component of the members of Vh is cont... |

7 |
Multigrid algorithms for mixed methods for second order elliptic problems
- Chen
- 1994
(Show Context)
Citation Context ... efficient iterative algorithms for solving the linear system arising from mixed finite element methods for linear and quasilinear second order elliptic problems in IR 2 and IR 3 . In the first paper =-=[11]-=-, a new approach for developing multigrid algorithms for the mixed finite element methods was introduced. It was first shown that the mixed finite element formulation can be algebraically condensed to... |

7 |
Domain decomposition and mixed nite element methods for elliptic problems
- Glowinski, Wheeler
(Show Context)
Citation Context ...domain decomposition, convergence, projection of coe cient. Partly supported by the Department of Energy under contract DE-ACOS-840R21400. 12 CHEN, EWING, AND LAZAROV limited to two space dimensions =-=[24, 25, 26, 27, 28, 31, 32, 40]-=-. The other method is the so-called dual variable method [19, 16, 18, 27, 28]. This approach makes use of a discretization of the ux operator (the coe cient times the gradient), which transfers the or... |

5 |
On the existence, uniqueness and convergence of nonlinear mixed finite element methods
- Chen
- 1989
(Show Context)
Citation Context ...( h ; v \DeltasE ) @En@\Omega = (g; v \Delta ) \Gamma 1 ; 8v 2 ~ V h ; (8.2b) X E2Eh (oe h \DeltasE ; ��) @En@\Omega = 0; 8�� 2 L h ; (8.2c) where ff h (u h ) = P h ff(u h ). It has a unique s=-=olution [8]-=-, [10], [12], [32]. A linearized version of (8.2) can be constructed as follows. Starting from any (oe 0 h ; u 0 h ;s0 h ) 2 ~ V h \Theta W h \Theta L h , we DOMAIN DECOMPOSITION ALGORITHMS FOR MIXED ... |

5 | Approximation of coe cients in hybrid and mixed methods for nonlinear parabolic problems - Chen, Douglas - 1991 |

5 |
Balancing domain decomposition for mixed nite elements
- Cowsar, Mandel, et al.
(Show Context)
Citation Context ...tment of Energy under contract DE-ACOS-840R21400. 12 CHEN, EWING, AND LAZAROV limited to two space dimensions [24, 25, 26, 27, 28, 31, 32, 40]. The other method is the so-called dual variable method =-=[19, 16, 18, 27, 28]-=-. This approach makes use of a discretization of the ux operator (the coe cient times the gradient), which transfers the original saddle point problem to an elliptic problem for the scalar unknown and... |

5 | Mixed nite element methods for quasilinear second order elliptic problems - Milner - 1985 |

3 |
Dual-variable Schwarz methods for mixed finite elements
- Cowsar
- 1993
(Show Context)
Citation Context ...ND LAZAROV divergence free vectors. This approach is limited to two space dimensions [22], [23], [24], [25], [26], [29], [30], [31], [39]. The other method is the so-called dual variable method [15], =-=[16]-=-, [17], [25], [26]. This approach makes use of a discretization of the flux operator (the coefficient times the gradient), which transfers the original saddle point problem to an elliptic problem for ... |