Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-05T07:57:48.363Z Has data issue: false hasContentIssue false
  • English
  • Français

A posteriori error estimates for linear exterior problemsvia mixed-FEM and DtN mappings

Published online by Cambridge University Press:  15 May 2002

Mauricio A. Barrientos ,
Gabriel N. Gatica  and
Matthias Maischak
Mauricio A. Barrientos
Affiliation:
GIMA, Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile. [email protected].
Gabriel N. Gatica
Affiliation:
GIMA, Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile. [email protected].
Matthias Maischak
Affiliation:
Institut für Angewandte Mathematik, Universität Hannover, Welfengarten 1, 30167 Hannover, Germany. [email protected].
Get access

Abstract

In this paper we combine the dual-mixed finite element method with a Dirichlet-to-Neumann mapping(given in terms of a boundary integral operator) to solve linear exterior transmission problems inthe plane. As a model we consider a second order elliptic equation in divergence form coupled withthe Laplace equation in the exterior unbounded region. We show that the resulting mixed variationalformulation and an associated discrete scheme using Raviart-Thomas spaces are well posed, and derivethe usual Cea error estimate and the corresponding rate of convergence. In addition, we develop twodifferent a-posteriori error analyses yielding explicit residual and implicit Bank-Weiser typereliable estimates, respectively. Several numerical results illustrate the suitability of theseestimators for the adaptive computation of the discrete solutions.

Keywords

Dirichlet-to-Neumann mappingmixed finite elementsRaviart-Tho mas spacesresidual based estimatesBank-Weiser approach.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 36 , Issue 2 , March 2002 , pp. 241 - 272
Copyright
© EDP Sciences, SMAI, 2002

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Ainsworth, M. and Oden, J.T., A unified approach to a posteriori error estimation using element residual methods. Numer. Math. 65 (1993) 23-50. CrossRef
I. Babuska and A.K. Aziz, Survey lectures on the mathematical foundations of the finite element method, in The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations, A.K. Aziz Ed., Academic Press, New York (1972).
Bank, R.E. and Weiser, A., Some a posteriori error estimators for elliptic partial differential equations. Math. Comp. 44 (1985) 283-301. CrossRef
M.A. Barrientos, A-posteriori Error Analysis of Dual-Mixed Variational Formulations for Linear and Nonlinear Boundary Value Problems (spanish). Ph.D. thesis, Universidad de Concepción, Concepción, Chile (in preparation).
M.A. Barrientos, G.N. Gatica and N. Heuer, An a-posteriori error estimate for a linear-nonlinear transmission problem in plane elastostatics. Technical Report 00-11, Departamento de Ingeniería Matemática, Universidad de Concepción (2000). Calcolo (to appear).
M.A. Barrientos, G.N. Gatica and E.P. Stephan, A mixed finite element method for nonlinear elasticity: two-fold saddle point approach and a-posteriori error estimate. Technical Report 99-25, Departamento de Ingeniería Matemática, Universidad de Concepción (1999). Numer. Math. (to appear).
C. Bernardi, Optimal finite-element interpolation on curved domains. SIAM J. Numer. Anal. 26 (1989) 1212-1240.
F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods. Springer-Verlag, Berlin, Heidelberg, New York (1991).
Brink, U., Carstensen, C. and Stein, E., Symmetric coupling of boundary elements and Raviart-Thomas-type mixed finite elements in elastostatics. Numer. Math. 75 (1996) 153-174. CrossRef
Carstensen, C., A posteriori error estimate for the symmetric coupling of finite elements and boundary elements. Computing 57 (1996) 301-322. CrossRef
Carstensen, C., An a-posteriori error estimate for a first-kind integral equation. Math. Comp. 66 (1997) 139-155. CrossRef
C. Carstensen and S.A. Funken, Coupling of mixed finite elements and boundary elements. IMA J. Numer. Anal. 20 (2000) 461-480.
Carstensen, C., Funken, S.A. and Stephan, E.P., On the adaptive coupling of FEM and BEM in 2-d-elasticity. Numer. Math. 77 (1997) 187-221. CrossRef
Carstensen, C. and Stephan, E.P., Adaptive coupling of boundary elements and finite elements. RAIRO Modél. Math. Anal. Numér. 29 (1995) 779-817. CrossRef
P. Clément, Approximation by finite element functions using local regularisation. RAIRO Anal. Numér. 9 (1975) 77-84.
Gatica, G.N., Combination of mixed finite element and Dirichlet-to-Neumann methods in nonlinear plane elasticity. Appl. Math. Lett. 10 (1997) 29-35. CrossRef
Gatica, G.N., An application of Babuska-Brezzi's theory to a class of variational problems. Appl. Anal. 75 (2000) 297-303. CrossRef
G.N. Gatica and N. Heuer, A dual-dual formulation for the coupling of mixed-FEM and BEM in hyperelasticity. SIAM J. Numer. Anal. 38 (2000) 380-400.
Gatica, G.N., Heuer, N. and Stephan, E.P., An implicit-explicit residual error estimator for the coupling of dual-mixed finite elements and boundary elements in elastostatics. Math. Methods Appl. Sci. 24 (2001) 179-191. 3.0.CO;2-M>CrossRef
Gatica, G.N. and Hsiao, G.C., The uncoupling of boundary integral and finite element methods for nonlinear boundary value problems. J. Math. Anal. Appl. 189 (1995) 442-461. CrossRef
Gatica, G.N. and Meddahi, S., An a-posteriori error estimate for the coupling of BEM and mixed-FEM. Numer. Funct. Anal. Optim. 20 (1999) 449-472. CrossRef
Gatica, G.N. and Meddahi, S., A dual-dual mixed formulation for nonlinear exterior transmission problems. Math. Comp. 70 (2001) 1461-1480. CrossRef
Gatica, G.N. and Stephan, E.P., A mixed-FEM formulation for nonlinear incompressible elasticity in the plane. Numer. Methods for Partial Differential Equations 18 (2002) 105-128. CrossRef
Gatica, G.N. and Wendland, W.L., Coupling of mixed finite elements and boundary elements for linear and nonlinear elliptic problems. Appl. Anal. 63 (1996) 39-75. CrossRef
G.N. Gatica and W.L. Wendland, Coupling of mixed finite elements and boundary elements for a hyperelastic interface problem. SIAM J. Numer. Anal. 34 (1997) 2335-2356.
D. Givoli, Numerical Methods for Problems in Infinite Domains. Elsevier Science Publishers B.V. (1992), Studies in Applied Mechanics 33.
P. Grisvard, Elliptic Problems in Non-Smooth Domains. Monographs and Studies in Mathematics, Vol. 24, Pitman (1985).
Han, H. and Bao, W., The artificial boundary conditions for incompressible materials on an unbounded domain. Numer. Math. 77 (1997) 347-363. CrossRef
Han, H. and The, X. Wu approximation of the exact boundary conditions at an artificial boundary for linear elastic equations and its application. Math. Comp. 59 (1992) 21-37.
G.C. Hsiao and S. Zhang, Optimal order multigrid methods for solving exterior boundary value problems. SIAM J. Numer. Anal. 31 (1994) 680-694.
R. Kress, Linear Integral Equations. Springer-Verlag (1989).
Meddahi, S., Valdés, J., Menéndez, O. and Pérez, P., On the coupling of boundary integral and mixed finite element methods. J. Comput. Appl. Math. 69 (1996) 113-124. CrossRef
P. Mund and E.P. Stephan, An adaptive two-level method for the coupling of nonlinear FEM-BEM equations. SIAM J. Numer. Anal. 36 (1999) 1001-1021.
J.E. Roberts and J.-M. Thomas, Mixed and Hybrid Methods, in Handbook of Numerical Analysis, P.G. Ciarlet and J.L. Lions Eds., Vol. II, Finite Element Methods (Part 1), North-Holland, Amsterdam (1991).
R. Verfürth, A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Wiley-Teubner, Chichester (1996).
A. Zenisek, Nonlinear Elliptic and Evolution Problems and their Finite Element Approximations. Academic Press, London (1990).