B. Wohlmuth, Hierarchical a posteriori error estimators for mortar finite element methods with Lagrange multipliers, Tech. Rep. 749, Courant Institute, New York University, 1997.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... been analyzed, see e.g. 2] However, since it may be technically cumbersome to eliminate the constraints imposed by the matching conditions and since fast solvers are by now available for mixed formulations, the analysis as a saddle point problem has recently attracted interest, see e.g. [1, 6, 11]. Moreover, on a principal level the inf sup condition is also often hidden in the analysis of mortar elements (braess num.ruhr uni bochum.de) Fakult at f ur Mathematik, Ruhr Universit at, 44780 Bochum, Germany y (dahmen igpm.rwth aachen.de) Institut f ur Geometrie und Praktische ....

....in actual computations with mortar elements. This discrepancy (gap) prohibits the use of Brezzi s theory with the standard Sobolev spaces and their norms. For a rigorous treatment one had to resort to nonstandard methods. One possibility is to introduce mesh dependent norms as done e.g. in [6, 9, 11]. Continuity, ellipticity, and the inf sup condition as required by Brezzi s theory are then available. Another concept can be found in [1, 12] where the analysis is performed in a two stage process. In a rst step merely the direct variables are estimated by the nonconforming theory. In the ....

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B. Wohlmuth, Hierarchical a posteriori error estimators for mortar nite element methods with Lagrange multipliers, SIAM J. Numer. Anal. 36 (1999), 1636-1658.

....) # H 1 2 00 (# kl ) The space of discrete multipliers is defined as M h : # #kl #S M kl,h . 14) The kernel of the restriction operator is V h : v h # X h : b(v h , h ) 0 for h # M h . 15) As already anounced in the introduction we will employ mesh dependent norms as in [AT95, Woh99b]. Setting #w# 1 2,h,#kl : h 1 2 #w# 0,#kl , let #v h # 2 1,h : #v h # 2 1,# # #kl #S #[v h ]# 2 1 2,h,#kl = #v h # 2 1,# # #kl #S h 1 #[v h ]# 2 0,#kl , 16) ## 2 1 2,h : # #kl #S ## 2 1 2,h,#kl = # #kl #S h## 2 0,#kl . 17) A MULTIGRID ....

B. Wohlmuth. Hierarchical a posteriori error estimators for mortar finite element methods with lagrange multipliers. SIAM J. Numer. Anal., 36:1636--1658, 1999.

....classifications. 65N55, 65N35, 65F10. 1. Introduction. The mortar method as a special domain decomposition methodology appears to be particularly attractive because different types of discretizations can be employed in different parts of the domain. It has been analysed in a series of papers [7, 8, 24] mainly in connection with second order elliptic boundary value problems of the form Gamma div a(x) grad u(x) f(x) in Omega ; a(x) u n = g(x) on Gamma N ae Omega ; 1.1) u = 0 on Gamma D : Omega n Gamma N ; where a(x) is a (sufficiently smooth) uniformly positive definite ....

.... 1=2 00 in this context has also been pointed out in the recent investigation [6] Our present considerations, in particular, a verification of the ellipticity in the discrete case lead us, however, to somewhat different conclusions namely to adjust the norms and to use mesh dependent norms as in [24]. Once a decision on the norms has been made, the analysis proceeds almost independently of the fact whether it is done in the framework of saddle point problems or of the theory of nonconforming elements. The results of Sections 2 and 3 will be used in Section 4 to estimate the convergence of the ....

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B. Wohlmuth, Hierarchical a posteriori error estimators for mortar finite element methods with Lagrange multipliers. SIAM J. Numer. Anal. (to appear)

....Leinen, Yserentant [40] the resulting continuous defect problem is first discretized and then localized. The second possibility follows Bank and Weiser [8] where the defect problem is first localized and then discretized. These concepts have been generalized to nonconforming discretizations, [58, 95], mixed Raviart Thomas discretizations, 1, 29, 57, 60] and the Stokes problem [9, 88] In this section, we present a hierarchical basis error estimator which is based on a defect correction in an appropriate higher order space, a hierarchical splitting, and, finally, localization techniques (cf. ....

B. Wohlmuth, Hierarchical a posteriori error estimators for mortar finite element methods with Lagrange multipliers, Tech. Rep. 749, Courant Institute, New York University, 1997.

.... mortar finite elements have been introduced by Bernardi, Maday and Patera [6, 7] and have been subsequently studied with regard to both the efficient iterative solution (cf. e.g. 1, 2, 4, 5, 8, 13, 14] and adaptive refinement by means of appropriate a posteriori error estimators (cf. e.g. [10, 11, 12, 17, 18, 19]) The mortar finite element method is based on a macro hybrid formulation of (1.1) 1.2) with respect to the decomposition of Omega as given by (1.5) This formulation involves the skeleton S of the decomposition which is defined as the union of all common edges Gamma ij of two adjacent ....

....; 1 i N ; we assume the existence of constants 0 i ; 1 i N , with being sufficiently small and independent of h i such that (S2) jjj u 2;i Gamma u i i jjj e i i ; 1 i N where jjj Delta jjj: a i ( Delta; Delta) 1 i N . Then, the following result can be verified (cf. [19] and [11, 12] Theorem 4.1. Let j be the hierarchical type error estimator as given by (3.7) Under the saturation assumptions (S1) and (S2) there exist constants 0 fl Gamma depending only on the shape regularity of T and on ff; ff; fi; fi in (1.3) 1.4) such that fl j jjj e jjj Gamma ....

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B.WOHLMUTH; Hierarchical a posteriori error estimators for mortar finite elements with Lagrange multipliers. Tech. Rep. 749, Courant Institute, New York University, 1997

....the nonconforming formulation or the equivalent saddle point problem. An estimator for both variables (u; or only for the weak solution u can be given. Here we follow the concept proposed in [19] and the a posteriori error estimator is based on the saddle point problem. We refer the reader to [21, 28, 44] for the hierarchical basis type error estimator obtained by solving Neumann boundary problems on the subdomains. In this case the boundary data is given by the discrete Lagrange multiplier and a measure for nonconformity at the interface is taken into account. It is known that the error jjju ....

....and a measure for nonconformity at the interface is taken into account. It is known that the error jjju Gamma u n jjj k Gamma n k L is of order O(h 2n ) if the weak solution u of (1. 1) and : n Delta aeru are regular enough, and the discrete Babu#ska Brezzi condition is satisfied [11, 13, 44]. Here, jjjvjjj : a (v; v) is the energy norm, and k Delta k L denotes the mesh dependent weighted L norm defined by L : 0;e ; v 2 L (S) 3.1) where ae e is the coefficient on the non mortar side. The weight ae e in the norm for the Lagrange multiplier shows that the ....

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B. Wohlmuth, Hierarchical a posteriori error estimators for mortar finite element methods with Lagrange multipliers. Preprint TR-749, Courant Institute of Mathematical Sciences. New York University, 1997.

.... done during the last few years; cf. e.g. 1, 4, 5, 14, 15] For the construction of efficient iterative solvers we refer to [2, 3, 20, 21] The concepts of a posteriori error estimators and adaptive refinement techniques have also been generalized to mortar methods on nonmatching grids; see e.g. [13, 22, 25, 24]. Originally introduced for the coupling of spectral element methods and finite elements, this method has thus now been extended to a variety of special situations [6, 7, 11, 12, 26] 1991 Mathematics Subject Classification. 65N15, 65N20. Key words and phrases. mortar finite elements, mixed ....

B. Wohlmuth, Hierarchical a posteriori error estimators for mortar finite element methods with lagrange multipliers, Tech. Report 749, Courant Institute of Math. Sciences, New York University, 1997, Submitted.

....as given by (4.28) Then, under the saturation assumptions (4.19) and (4.29) there exist constants 0 fl H Gamma H depending only on the shape regularity of T 0 and on ff; ff; fi; fi in (2.3) 2.4) such that fl H j H jjj e jjj Gamma H j H : 4.30) Proof. For the proof we refer to [39]. ffl Remark 4.2. In practice, we do not compute (u M ; M ) 2 Q N i=1 S 1( Omega i ; T (k) i ) Theta M 1 (S) exactly but only some approximation ( u M ; M ) by means of an iterative solution process as, e.g. that one described in the previous section. In this case, the iteration ....

....with the smaller diffusion coefficient along with a sharp resolution of the interfaces. This is obviously in contrast to the use of standard conforming meshes where, due to the continuity requirements, we also encounter strong refinement in the other regions close to the interfaces. We refer to [38, 39] for further numerical results. Tables 5.1a and 5.1b contain the history of the refinement process displaying the number DDM on Nonmatching Grids 17 of unknowns, the estimated and the true error, as well as the effectivity index per level of refinement. Note that the effectivity index is the ....

B.WOHLMUTH; Hierarchical a posteriori error estimators for mortar finite element methods with Lagrange multipliers. Tech. Report, Courant Institute of Math. Sciences, New York, 1997

....the L 2 and H 1 norm as well as for the Lagrange multiplier in the H 1=2 00 dual norm and a weighted L 2 norm are obtained. For the standard approach, we refer to [Ben95, BM95, BMP93, BMP94] The introduction of the mesh dependent norm for the Lagrange multiplier can be found in [Woh97] and the a priori estimates in case of a dual basis in [Woh98] Let us first give the problem setting. We consider the model problem Lu : Gammadiv (aru) b u = f in Omega ; u = 0 on Gamma : Omega ; 2.1) where Omega is a bounded, polygonal domain in IR 2 and f 2 L 2( Omega Gamma3 ....

....a H 1=2 00 dual norm or a mesh dependent L 2 norm kk 2 M : M X m=1 kk 2 (H 1=2 00 (fl m ) 0 ; or kk 2 M : M X m=1 X e2 Sigma m h e kk 2 0;e : In case of (u i h ; i h ) and i = 1 or i = 2, the a priori estimates (2.5) and (2. 6) are well known [Ben95, BMP93, BMP94, Woh97, Woh98] For the cases i = 3 and i = 4, they can be found in [Woh99a] We refer to [Woh98, Woh99a] for a numerical comparison of the discretization errors. It is shown that for examples including interior subdomains, singularities or discontinuous coefficients not only the asymptotic rates are ....

B. Wohlmuth. Hierarchical a posteriori error estimators for mortar finite element methods with lagrange multipliers. Preprint 749, Courant Institute of Mathematical Sciences, New York University, 1997. to appear in SIAM J. Numer. Anal.

....Leinen, Yserentant [40] the resulting continuous defect problem is first discretized and then localized. The second possibility follows Bank and Weiser [8] where the defect problem is first localized and then discretized. These concepts have been generalized to nonconforming discretizations, [58, 95], mixed Raviart Thomas discretizations, 1, 29, 57, 60] and the Stokes problem [9, 88] In this section, we present a hierarchical basis error estimator which is based on a defect correction in an appropriate higher order space, a hierarchical splitting, and, finally, localization techniques (cf. ....

B. Wohlmuth, Hierarchical a posteriori error estimators for mortar finite element methods with Lagrange multipliers, Tech. Rep. 749, Courant Institute, New York University, 1997.

....The following saddle point problem is equivalent to (2.2) Find (u h ; h ) 2 X h Theta M h such that a(u h ; v h ) b(v h ; h ) f(v h ) v h 2 X h b(u h ; h ) 0; h 2 M h : 2. 3) The Lagrange multiplier h is an approximation of the flux at the interfaces : a u n , see [5, 29]. We remark that the sign in the definition of the jump does not influence the solution u h and the orientation of n depends on the definition of the jump. In the rest of this section, we state technical tools which will be necessary for the analysis of the multigrid method. In particular, the ....

....approach will be based on the saddle point formulation (2.3) It is introduced and analyzed in [5] where a discrete inf sup condition is given in the H 1=2 00 norm of the Lagrange multiplier. Here, we work with mesh dependent norms for which a discrete inf sup condition is established in [29] jjjj 2 h Gammas ;S : X fl m X e2Sm;h h 2s e jjjj 2 0;e ; 2 M h ; s 0 see also [10, 11] The notation jj Delta jj h Gammas ;S is choosen because the mesh dependent norm represents a kind of discrete H s dual norm. Then, the following inf sup condition holds true cjj h jj h ....

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B.I. Wohlmuth, Hierarchical a posteriori error estimators for mortar finite element methods with Lagrange multipliers. to appear in SIAM J. Numer. Anal.

....that a unique solution of (2.2) exists and that the discretization error is of order h if the solution of (2.1) is smooth; see [6, 7] In a second, equivalent approach the space f M h explicitly plays the role of a Lagrange multiplier space. This approach is studied in [4] and used further in [11, 25, 26]. The resulting variational formulation gives rise to a saddle point problem: Find ( u h ; h ) 2 X h Theta f M h such that a( u h ; v h ) b( h ; v h ) f(v h ) v h 2 X h ; b( h ; u h ) 0; h 2 f M h : 2.3) In particular, it can be easily seen that the first component of the ....

....to consider a priori estimates for aru Delta n lk Gamma h in suitable norms. Here n lk is the outer unit normal of Omega k restricted to Gamma lk . This issue was first addressed in [4] where a priori estimates in the (H 1=2 00 ) 0 norm were established. Similar bounds are given in [26] for a weighted L 2 norm. As in the general saddle point approach [13] the essential point is to establish adequate inf sup conditions; such bounds have been established with constants independent of h for both these norms; see [4, 26] In the following, all constants 0 c C 1 are generic ....

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B.I. Wohlmuth, Hierarchical a posteriori error estimators for mortar finite element methods with Lagrange multipliers. to appear in SIAM J. Numer. Anal.

....results, we obtain: Theorem 3.1. Under the saturation assumptions (3.17) and (3.20) there holds c jjj e jjj 2 Omega N X i=1 jjj e i jjj 2 Omega i L X l=1 X E2 Gamma l h Gamma1 E k[u m ] J k 2 0;E C jjj e jjj 2 Omega (3.22) For a proof of Theorem 3.1. we refer to [31]. A localization of the Neumann problems (3.16) can be achieved by taking advantage of the hierarchical two level splitting S 2( Omega i ; T k) i ) S 1( Omega i ; T (k) i ) Phi S 2( Omega i ; T (k) i ) 3.23) where S 2( Omega i ; T (k) i ) stands for the hierarchical ....

....Remark 3.3. We note that residual based a posteriori error estimators for mortar finite element discretizations have been developed in [16, 30] The hierarchical basis error estimator and a fully hierarchical basis error estimator that includes the error (n Delta aru) have been analyzed in [31]. 4. Numerical results In this section, we present some numerical results illustrating the benefits of the adaptive mixed and the adaptive macro hybrid finite element methods. As an example for the mixed hybrid approach, we consider (1.1) with a = 1:0 and b = 0:25 and homogeneous Dirichlet ....

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B.WOHLMUTH; Hierarchical a posteriori error estimators for mortar finite element methods with Lagrange multipliers. Tech. Report, Courant Institute of Math. Sciences, New York, 1997

.... done during the last few years; cf. e.g. 1, 4, 5, 14, 15] For the construction of efficient iterative solvers we refer to [2, 3, 19, 20] The concepts of a posteriori error estimators and adaptive refinement techniques have also been generalized to mortar methods on nonmatching grids; see e.g. [13, 21, 23, 24]. Originally introduced for the coupling of spectral element methods and finite elements, this method has thus now been extended to a variety of special situations [6, 7, 11, 12, 25] 2 The continuous problem We consider the following elliptic boundary value problem Lu : Gammadiv (aru) b u = ....

B. Wohlmuth. Hierarchical a posteriori error estimators for mortar finite element methods with Lagrange multipliers. Report, Courant Institute of Math. Sciences, New York, 1997.

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