# A finite element method for domain decomposition with non-matching grids

Roland Becker; Peter Hansbo; Rolf Stenberg

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

- Volume: 37, Issue: 2, page 209-225
- ISSN: 0764-583X

## Access Full Article

top## Abstract

top## How to cite

topBecker, Roland, Hansbo, Peter, and Stenberg, Rolf. "A finite element method for domain decomposition with non-matching grids." ESAIM: Mathematical Modelling and Numerical Analysis 37.2 (2010): 209-225. <http://eudml.org/doc/194159>.

@article{Becker2010,

abstract = {
In this note, we propose and analyse a method for handling
interfaces between non-matching grids based on an approach
suggested by Nitsche (1971) for the approximation of
Dirichlet boundary conditions. The exposition is limited to
self-adjoint elliptic problems, using Poisson's equation as a
model. A priori and a posteriori error estimates are given. Some
numerical results are included.
},

author = {Becker, Roland, Hansbo, Peter, Stenberg, Rolf},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis},

keywords = {Nitsche's method; domain decomposition; non-matching grids.; non-matching grids; Poisson problem; error estimates; numerical results; finite element method},

language = {eng},

month = {3},

number = {2},

pages = {209-225},

publisher = {EDP Sciences},

title = {A finite element method for domain decomposition with non-matching grids},

url = {http://eudml.org/doc/194159},

volume = {37},

year = {2010},

}

TY - JOUR

AU - Becker, Roland

AU - Hansbo, Peter

AU - Stenberg, Rolf

TI - A finite element method for domain decomposition with non-matching grids

JO - ESAIM: Mathematical Modelling and Numerical Analysis

DA - 2010/3//

PB - EDP Sciences

VL - 37

IS - 2

SP - 209

EP - 225

AB -
In this note, we propose and analyse a method for handling
interfaces between non-matching grids based on an approach
suggested by Nitsche (1971) for the approximation of
Dirichlet boundary conditions. The exposition is limited to
self-adjoint elliptic problems, using Poisson's equation as a
model. A priori and a posteriori error estimates are given. Some
numerical results are included.

LA - eng

KW - Nitsche's method; domain decomposition; non-matching grids.; non-matching grids; Poisson problem; error estimates; numerical results; finite element method

UR - http://eudml.org/doc/194159

ER -

## References

top- J.-P. Aubin, Approximation of Elliptic Boundary-Value Problem. Wiley (1972).
- D. Arnold, An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal.19 (1982) 742-760.
- C. Baiocchi, F. Brezzi and L.D. Marini, Stabilization of Galerkin methods and applications to domain decomposition, in Future Tendencies in Computer Science, Control and Applied Mathematics, A. Bensoussan and J.-P. Verjus Eds., Springer (1992) 345-355.
- J.C. Barbosa and T.J.R. Hughes, Boundary Lagrange multipliers in finite element methods: error analysis in natural norms. Numer. Math.62 (1992) 1-15.
- J.W. Barrett and C.M. Elliot, Finite element approximation of the Dirichlet problem using the boundary penalty method. Numer. Math.49 (1986) 343-366.
- R. Becker and P. Hansbo, Discontinuous Galerkin methods for convection-diffusion problems with arbitrary Péclet number, in Numerical Mathematics and Advanced Applications: Proceedings of the 3rd European Conference, P. Neittaanmäki, T. Tiihonen and P. Tarvainen Eds., World Scientific (2000) 100-109.
- R. Becker and R. Rannacher, A feed-back approach to error control in finite element methods: basic analysis and examples. East-West J. Numer. Math.4 (1996) 237-264.
- C. Bernadi, Y. Maday and A. Patera, A new nonconforming approach to domain decomposition: the mortar element method, in Nonlinear Partial Differential Equations and Their Application, H. Brezis and J.L. Lions Eds., Pitman (1989).
- F. Brezzi, L.P. Franca, D. Marini and A. Russo, Stabilization techniques for domain decomposition methods with non-matching grids, IAN-CNR Report N. 1037, Istituto di Analisi Numerica Pavia.
- J. Freund and R. Stenberg, On weakly imposed boundary conditions for second order problems, in Proceedings of the Ninth Int. Conf. Finite Elements in Fluids, M. Morandi Cecchi et al. Eds., Venice (1995) 327-336.
- J. Freund, Space-time finite element methods for second order problems: an algorithmic approach. Acta Polytech. Scand. Math. Comput. Manage. Eng. Ser. 79 (1996).
- B. Heinrich and S. Nicaise, Nitsche mortar finite element method for transmission problems with singularities. SFB393-Preprint 2001-10, Technische Universität Chemnitz (2001).
- B. Heinrich and K. Pietsch, Nitsche type mortaring for some elliptic problem with corner singularities. Computing68 (2002) 217-238.
- C. Johnson and P. Hansbo, Adaptive finite element methods in computational mechanics. Comput. Methods Appl. Mech. Engrg.101 (1992) 143-181.
- P. Le Tallec and T. Sassi, Domain decomposition with nonmatching grids: augmented Lagrangian approach. Math. Comp.64 (1995) 1367-1396.
- P.L. Lions, On the Schwarz alternating method III: a variant for nonoverlapping subdomains, in Third International Symposium on Domain Decomposition Methods for Partial Differential Equations, T.F. Chan, R. Glowinski, J. Periaux and O.B. Widlund Eds., SIAM (1989) 202-223.
- J. Nitsche, Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind. Abh. Math. Sem. Univ. Hamburg36 (1971) 9-15.
- R. Stenberg, On some techniques for approximating boundary conditions in the finite element method. J. Comput. Appl. Math.63 (1995) 139-148.
- R. Stenberg, Mortaring by a method of J.A. Nitsche, in Computational Mechanics: New Trends and Applications, S. Idelsohn, E. Onate and E. Dvorkin Eds., CIMNE, Barcelona (1998).
- V. Thomée, Galerkin Finite Element Methods for Parabolic Problems. Springer (1997).
- B.I. Wohlmuth, A residual based error estimator for mortar finite element discretizations. Numer. Math.84 (1999) 143-171.

## Citations in EuDML Documents

top- Christian Grossmann, Penalties, Lagrange multipliers and Nitsche mortaring
- Tie Zhu Zhang, Shu Hua Zhang, Optimal convergence and a posteriori error analysis of the original DG method for advection-reaction equations
- Saber Amdouni, Patrick Hild, Vanessa Lleras, Maher Moakher, Yves Renard, A stabilized Lagrange multiplier method for the enriched finite-element approximation of contact problems of cracked elastic bodies
- Saber Amdouni, Patrick Hild, Vanessa Lleras, Maher Moakher, Yves Renard, A stabilized Lagrange multiplier method for the enriched finite-element approximation of contact problems of cracked elastic bodies
- Anita Hansbo, Peter Hansbo, Mats G. Larson, A finite element method on composite grids based on Nitsche’s method
- Anita Hansbo, Peter Hansbo, Mats G. Larson, A finite element method on composite grids based on Nitsche's method
- Saber Amdouni, Patrick Hild, Vanessa Lleras, Maher Moakher, Yves Renard, A stabilized Lagrange multiplier method for the enriched finite-element approximation of contact problems of cracked elastic bodies
- Carlo D’Angelo, Anna Scotti, A mixed finite element method for Darcy flow in fractured porous media with non-matching grids
- Carlo D’Angelo, Anna Scotti, A mixed finite element method for Darcy flow in fractured porous media with non-matching grids
- V. Lleras, A Stabilized Lagrange Multiplier Method for the Finite Element Approximation of Frictional Contact Problems in Elastostatics

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.