D. N. Arnold and F. Brezzi. Mixed and nonconforming nite element methods: Implementation, postprocessing and error estimates. RAIRO, Mod. Math. Anal. Numer. 19:7-32, 1985.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....P and S to the discrete analogues of (16) and (17) are globally polynomials, a totally uninteresting case. In particular, for the lowest index Raviart Thomas space to be employed in this paper, these functions would have to be globally constant. So, let us introduce Lagrange multipliers [3, 30] jk ( g for the global pressure system (16) and = s for the saturation system (17) on the edges f jk g. Assume that, when Q j = Q j , 2 V , its normal component Q n j on jk is a polynomial of some xed degree = degree(Q n j ) where for simplicity we shall assume to ....

D. N. Arnold and F. Brezzi, Mixed and nonconforming nite element methods: implementation, postprocessing and error estimates, R.A.I.R.O. Modelisation Mathematique et Analyse Numerique, 19 (1985) 7-32.

....assumption must be monitored during AMR. Also note that Peaceman s model was developed for block centered nite di erence discretization of the pressure equation, which is known [90] to be equivalent to the lowest order mixed nite element method with appropriate quadrature, which is in turn known [5] to produce the same pressure and velocity eld as the hybrid mixed nite element method used in this paper. For extension of the areal model to quadrilateral grids or three dimensions, see [69, 77] 2.2.2.3 Boundary Conditions in Two Dimensions If we specify the ow at the boundary, then in a ....

....system of linear equations for the Lagrange multipliers. In this application, the Lagrange multipliers can be identi ed with uid pressures at the cell sides. 5. 1 Mathematical Formulation of the Hybrid Mixed Finite Element Method The weak formulation of the hybrid mixed method equations [5, 22, 23, 32, 81] is similar to, but less commonly used for porous ow than the mixed method [4, 5, 15, 22, 23, 24, 25, 44, 46, 47, 55, 56, 81] Let the problem domain = i;j be a union of rectangles, and let E be the set of sides of these rectangles interior to We want to nd v 2 i;j ) p 2 H i;j ) ....

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D.N. Arnold and F. Brezzi. Mixed and nonconforming nite element methods: Implementation, postprocessing and error estimates. RAIRO Model. Math. Anal. Numer., 19:7-32, 1985.

....nonconforming method in this part of the paper. Occasional trivial modi cations in the presentation suce to cover the simplicial case. We shall treat only decomposition into individual elements here; as in earlier work [7, 8] utilizing Robin transmission conditions, we begin by hybridizing [2] the nite element method. First note that v= jk is constant on jk for any v 2 j ) Thus, it is reasonable to de ne a hybridization of (3.2) by associating a space of Lagrange multipliers e 2 associated with a( jk ) p= jk on jk . Also, localize the nonconforming Galerkin space ....

D. N. Arnold and F. Brezzi. Mixed and nonconforming nite element methods: implementation, postprocessing and error estimates. R.A.I.R.O. Modelisation Mathematique et Analyse Numerique, 19 (1985) 7-32.

....saturation calculation is very similar to that described in complete detail in [23] the only di erence is in the de nition of the saturation ux V . Its de nition in this paper leads to a simpler procedure in the presence of gravity terms. Otherwise, the implementation can be based on hybridizing [5] the mixed method by introducing Lagrange multipliers for the saturation on the faces of the elements and applying a standard quadrature based on the vertices before introducing a domain decomposition iteration based on Robin transmission conditions to solve the algebraic equations. This technique ....

D. N. Arnold and F. Brezzi, Mixed and nonconforming nite element methods: implementation, postprocessing and error estimates, R.A.I.R.O. Modelisation Mathematique et Analyse Numerique, 19 (1985) 7-32.

....problem. A somewhat more complicated argument would have eliminated the need to require K i 0; see [11] Therefore, Theorem 3.1 holds also for purely real K and . 4 The Hybridized Procedure To hybridize the nonconforming procedure in the manner of Fraeijs de Veubeke [13] and Arnold Brezzi [3], we employ e h as a space of Lagrange multipliers, associating elements h 2 e h with 1 u h jk ( jk ) on jk . We also localize the space NC h by introducing the new space NC h 1 = v 2 L 2( vj j 2 NC h j : The hybridized nonconforming procedure then ....

D. N. Arnold and F. Brezzi. Mixed and nonconforming nite element methods: implementation, postprocessing and error estimates. R.A.I.R.O. Modelisation Mathematique et Analyse Numerique, 19 (1985), pp. 7-32.

....h k 2 0 C 2 sup X T2T k h 2 T ka 1 (j j h )k 2 div ;T (3.12) where the constant C sup only depends on the geometry of the initial triangulation and on the ratio of the local bounds of the coecients in (1. 1) In case of the Poisson equation, this result is well established (cf. e.g. [2, 12]) In the general case, it can be proved assuming a discrete H 2 regularity and using some duality techniques (cf. e.g. 12; Remark 2.16] Lemma 3.6 There exist constants c ue , c j e , C ue , C j e , C 1 it 0 independent of the re nement level such that c ue ku e k 0 jja 1 h ....

....g and B( T k ) fv 2 L 2( j vj T 2 P 3 (T ) vj T = 0; T 2 T k g. The local operator 0 is the L 2 projection onto W 0( T k ) whereas P a 1 denotes the orthogonal projection onto RT 1 0( T k ) with respect to the bilinear form a ( Then, the following equalities hold true [2, 12] u h = 0 NC ; h = e NC ; j h = P a 1 (ar NC ) 6.2) where (j h ; u h ; h ) 2 RT 1 0( T k ) W 0( T k ) M 0( E k ) denotes the unique solution of the mixed hybrid formulation of the variational problem (1.4) a (j h ; q) b (q; u h ) d ( h ; q) 0; q 2 RT 1 0( T k ) ....

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D. Arnold and F. Brezzi, Mixed and nonconforming nite element methods: Implementation, post{ processing and error estimates, M 2 AN Math. Modelling Numer. Anal., 19 (1985), pp. 7-35.

.... by Aj I i = A i def = 1 h i Z x i x i 1 e V (s) T ds; i = 1; N; Bj I i = B i def = e Vmin =T ; i = 1; N: J h 2 is an approximation of the energy ux J 2 , g h 2 is a piecewise constant approximation of g 2 and g h 2 is an approximation of g 2 at the nodes (see [3, 22]) The rst equation is obtained from a weak version of (39) using integration by parts and summation over all I i together with the inverse of the Slotboom transformation (38) Notice that the discrete inverse transformation is not the same for the variables g h 2 and g h 2 . We refer to ....

D. N. Arnold and F. Brezzi. Mixed and nonconforming nite element methods: Implementation, postprocessing and error estimates. RAIRO, Mod. Math. Anal. Numer. 19:7-32, 1985.

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D. N. Arnold and F. Brezzi. Mixed and nonconforming nite element methods: Implementation, postprocessing and error estimates. RAIRO, Mod. Math. Anal. Numer. 19:7-32, 1985.

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Arnold, D. N. and Brezzi F.: \Mixed and nonconforming - nite element methods: Implementation, postprocessing and error estimates," Math. Anal. Num, 19:7-32, 1985.

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Arnold DN, Brezzi F. Mixed and nonconforming #nite element methods: implementation, postprocessing and error estimates. RAIRO Mod# elisation Mathematique et Analyse Num# erique 1985; 19:7 -- 32.

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D. N. Arnold and F. Brezzi. Mixed and nonconforming nite element methods: implementation, postprocessing and error estimates. R.A.I.R.O. Modelisation Mathematique et Analyse Numerique, 19 (1985) 7-32.

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D. N. Arnold and F. Brezzi, Mixed and nonconforming nite element methods: implementation, postprocessing and error estimates, R.A.I.R.O. Modelisation Mathematique et Analyse Numerique, 19 (1985), pp. 7-32.

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D. N. Arnold and F. Brezzi. Mixed and nonconforming nite element methods: implementation, postprocessing and error estimates. R.A.I.R.O. Modelisation Mathematique et Analyse Numerique, 19 (1985) 7-32.

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