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A Least-Squares/Fictitious Domain Method for Linear Elliptic Problems with Robin Boundary Conditions

Published online by Cambridge University Press:  20 August 2015

Roland Glowinski  and
Qiaolin He
Roland Glowinski*
Affiliation:
Department of Mathematics, University of Houston, Houston, TX 77204, USA Institute of Advanced Study, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong
Qiaolin He*
Affiliation:
Department of Mathematics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong Department of Mathematics, Sichuan University, Chengdu 610064, China
*
Email:[email protected]
Corresponding author.Email:[email protected]
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Abstract

In this article, we discuss a least-squares/fictitious domain method for the solution of linear elliptic boundary value problems with Robin boundary conditions. Let Ω and ω be two bounded domains of Rd such that ω̅⊂Ω. For a linear elliptic problem in Ω\ω̅ with Robin boundary condition on the boundary ϒ of ω, our goal here is to develop a fictitious domain method where one solves a variant of the original problem on the full Ω, followed by a well-chosen correction over ω. This method is of the virtual control type and relies on a least-squares formulation making the problem solvable by a conjugate gradient algorithm operating in a well chosen control space. Numerical results obtained when applying our method to the solution of two-dimensional elliptic and parabolic problems are given; they suggest optimal order of convergence.

Keywords

65M8565N8576M1093E24Least-Square methodsfictitious domain methodsfinite element methodsRobin boundary conditions
Type
Research Article
Information
Communications in Computational Physics , Volume 9 , Issue 3 , March 2011 , pp. 587 - 606
Copyright
Copyright © Global Science Press Limited 2011

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References

[1]Hyman, M. A., Non-iterative numerical solution of boundary value problems, Appl. Sci. Res. Sec., B2 (1952), 325–351.Google Scholar
[2]Saul’ev, V. K., On a method for automatization of solution of boundary value problems on high performance computers, Dokl. Acad. Sci. USSB., 144(3) (1962), 497–500.Google Scholar
[3]Saul’ev, V. K., On the solution of some boundary value problems on high performance computers by fictitious domain methods, Siberian J. Math., 4(4) (1963), 912–925.Google Scholar
[4]Buzbee, B. L., Dorr, F. W., George, J. A. and Golub, G. H., The direct numerical solution of the discrete Poisson equation on irregular regions, SIAM J. Numer. Anal., 8 (1971), 722–736.Google Scholar
[5]Glowinski, R., Pan, T. W. and Periaux, J., A fictitious domain method for Dirichlet problems and applications, Comput. Methods. Appl. Mech. Eng., 111 (1994), 283–303.Google Scholar
[6]Glowinski, R., Pan, T. W. and Periaux, J., A fictitious domain method for external incompressible flow modeled by Navier-Stokes equations, Comput. Methods. Appl. Mech. Eng., 112 (1994), 133–148.CrossRefGoogle Scholar
[7]Glowinski, R., Pan, T. W. and Periaux, J., A Lagrange multiplier/fictitious domain method for the Dirichlet problem, generalization to some flow problems, Japan J. Indust. Appl. Math., 12 (1995), 87–108.Google Scholar
[8]Glowinski, R., Pan, T. W., Kearsley, A. J. and Periaux, J., Numerical simulation and optimal shape for viscous flow by a fictitious domain method, Int. J. Numer. Methods. Fluids., 20 (1995), 695–711.Google Scholar
[9]Glowinski, R., Pan, T. W., Hesla, T. and Joseph, D. D, A distributed Lagrange multiplier/fictitious method for flows around moving rigid bodies: application to particulate flows, Int. J. Multiphase. Flow., 25 (1999), 755–794.Google Scholar
[10]Pan, T. W., Glowinski, R. and Galdi, G. P., Direct simulation of the motion of settling ellipsoid in Newtonian fluid, J. Comp. Appl. Math., 149 (2002), 71–82.Google Scholar
[11]Lions, J. L., Virtual and effective control for distributed systems and decomposition of everything, Journal d’Analyse Mathématique, 80(1) (2000), 257–297.CrossRefGoogle Scholar
[12]Glowinski, R., Finite element method for incompressible viscous flow, In Handbook of Numerical Analysis, Vol. IX, Ciarlet, P. G. and Lions, J. L., eds., North-Holland, Amsterdam, 3–1176, 2003.Google Scholar
[13]Glowinski, R., Lions, J. L. and He, J., Exact and Approximate Controllability for Distributed Parameter Systems: A Numerical Approach, Cambridge University Press, Cambridge, UK, 2008.Google Scholar