Partial Differential Equations - Modelling and ... - ResearchGate
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40 Yu.A. Kuznetsov The matrix M 0 in (67) is obtained by the assembling of matrices M 0,(k) H defined in (36) if the coefficient c is a positive function in E k or in (58) if c ≡ 0 in E k , k = 1,n. Respectively, the components ˆp k of the vector ¯p H in (67) are defined either in (9) if the coefficient c is a positive function in E k or in (65) if c ≡ 0inE k , k = 1,n. The elimination of ū H (condensation of the system (67)) results in the algebraic system in terms of vector ¯p H and the interface and boundary Lagrange multiplier vector ¯λ: ] A =¯q. (68) [¯pH¯λ Here, where A k = [ ] ck |E k | 0 + 0 0 A = n∑ N k A k Nk T , (69) k=1 [ B 0,(k) H C k ] [ M 0,(k) H ] −1 [ ( B 0,(k) H ) T C T k ] (70) are symmetric and positive definite matrices, and N k are the underlying assembling matrices, k = 1,n. The formula for the vector ¯q in (68) can be easily derived. Remark 1. If the function f is constant in E ≡ E k then the vector F in (23) is defined by the formula F = −f E Dē, (71) where f E is the value of f in E, and belongs to the null space of the matrix S + in (51). To this end, instead of (54) we have ū h = R 1 ū H , (72) and ḡ in (57) is the zero vector. Simple analysis shows that the resulting discretization (66) is equivalent to the “div-const” discretization proposed in [KR03] (see also [KLS04, KR05]). Remark 2. The previous remark is concerned the case when c ≡ 0inE ≡ E k , 1 ≤ k ≤ n. Consider the case when c is a positive function in E, the diffusion equation is discretized by the method (16)–(18) and the value n k for this cell is equal to one. Under the assumptions made, the equation (index k is omitted) B H ū H + B h ū h − ˜Σ ¯p = F 1 , (73) where the matrix ˜Σ and the vector F 1 are defined in (20) and (21), respectively, is the underlying counterpart of the third equation in (23). Similar to (50), we can consider the following formula for the solution subvector ¯p: ¯p = S + [ h ˜BH ū H − ˜Σ ] ¯p − F 1 + αē (74)
with some coefficient α ∈ R where Mixed FE Methods on Polyhedral Meshes 41 S h = B h M −1 h BT h (75) and S + h is defined in (51). The vectors ˜Σ ¯p and F 1 belong to ker S + h . Therefore, instead of (74) we get ¯p = S + ˜B h H ū H + αē. (76) It proves that for the discretization method (16)–(18) the formula (72) is still valid, and the final discretization (66) is equivalent to the “div-const” discretization in [KR03]. Acknowledgement. This research was supported by Los Alamos Computational Sciences Institute (LACSI) and by ExxonMobil Upstream Research Company. The author is grateful to S. Maliassov and M. Shashkov for fruitful discussions, as well as to O. Boyarkin, V. Gvozdev, and D. Svyatskiy for numerical implementation and applications of the proposed method. References [BF91] F. Brezzi and M. Fortin. Mixed and hybrid finite element methods. Springer- Verlag, Berlin 1991 [Kuz05] Yu. Kuznetsov. Mixed finite element method in domains of complex geometry. In Abstract Book – 1st International Seminar of SCOMA, number A4/2005 in Reports of the Department of Mathematical Information Technology, Series A, Collections, University of Jyväskylä, Jyväskylä, 2005. [Kuz06] Yu. Kuznetsov. Mixed finite element method for diffusion equations on polygonal meshes with mixed cells. J. Numer. Math., 14(4):305–315, 2006 [KLS04] Yu. Kuznetsov, K. Lipnikov, and M. Shashkov. The mimetic finite difference method on polygonal meshes for diffusion-type equations. Comput. Geosci., 8:301–324, 2004 [KR03] Yu. Kuznetsov and S. Repin. New mixed finite element method on polygonal and polyhedral meshes. Russian J. Numer. Anal. Math. Modelling, 18(3):261–278, 2003 [KR05] Yu. Kuznetsov and S. Repin. Convergence analysis and error estimates for mixed finite element method on distorted meshes. J. Numer. Math., 13(1):33–51, 2005 [RT91] J. E. Roberts and J.-M. Thomas. Mixed and hybrid methods. In P.- G. Ciarlet and J.-L. Lions, editors, Handbook of Numerical Analysis, Vol. II, pages 523–639. North-Holland, Amsterdam, 1991.
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with some coefficient α ∈ R where<br />
Mixed FE Methods on Polyhedral Meshes 41<br />
S h = B h M −1<br />
h BT h (75)<br />
<strong>and</strong> S + h is defined in (51). The vectors ˜Σ ¯p <strong>and</strong> F 1 belong to ker S + h . Therefore,<br />
instead of (74) we get<br />
¯p = S + ˜B h H ū H + αē. (76)<br />
It proves that for the discretization method (16)–(18) the formula (72) is<br />
still valid, <strong>and</strong> the final discretization (66) is equivalent to the “div-const”<br />
discretization in [KR03].<br />
Acknowledgement. This research was supported by Los Alamos Computational Sciences<br />
Institute (LACSI) <strong>and</strong> by ExxonMobil Upstream Research Company. The<br />
author is grateful to S. Maliassov <strong>and</strong> M. Shashkov for fruitful discussions, as well<br />
as to O. Boyarkin, V. Gvozdev, <strong>and</strong> D. Svyatskiy for numerical implementation <strong>and</strong><br />
applications of the proposed method.<br />
References<br />
[BF91] F. Brezzi <strong>and</strong> M. Fortin. Mixed <strong>and</strong> hybrid finite element methods. Springer-<br />
Verlag, Berlin 1991<br />
[Kuz05] Yu. Kuznetsov. Mixed finite element method in domains of complex geometry.<br />
In Abstract Book – 1st International Seminar of SCOMA, number<br />
A4/2005 in Reports of the Department of Mathematical Information Technology,<br />
Series A, Collections, University of Jyväskylä, Jyväskylä, 2005.<br />
[Kuz06] Yu. Kuznetsov. Mixed finite element method for diffusion equations on<br />
polygonal meshes with mixed cells. J. Numer. Math., 14(4):305–315, 2006<br />
[KLS04] Yu. Kuznetsov, K. Lipnikov, <strong>and</strong> M. Shashkov. The mimetic finite difference<br />
method on polygonal meshes for diffusion-type equations. Comput.<br />
Geosci., 8:301–324, 2004<br />
[KR03]<br />
Yu. Kuznetsov <strong>and</strong> S. Repin. New mixed finite element method on polygonal<br />
<strong>and</strong> polyhedral meshes. Russian J. Numer. Anal. Math. <strong>Modelling</strong>,<br />
18(3):261–278, 2003<br />
[KR05] Yu. Kuznetsov <strong>and</strong> S. Repin. Convergence analysis <strong>and</strong> error estimates<br />
for mixed finite element method on distorted meshes. J. Numer. Math.,<br />
13(1):33–51, 2005<br />
[RT91] J. E. Roberts <strong>and</strong> J.-M. Thomas. Mixed <strong>and</strong> hybrid methods. In P.-<br />
G. Ciarlet <strong>and</strong> J.-L. Lions, editors, H<strong>and</strong>book of Numerical Analysis, Vol. II,<br />
pages 523–639. North-Holl<strong>and</strong>, Amsterdam, 1991.
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