American Institute of Mathematical Sciences

August  2008, 2(3): 317-333. doi: 10.3934/ipi.2008.2.317

An alternating boundary integral based method for a Cauchy problem for the Laplace equation in semi-infinite regions

 1 Faculty of Applied Mathematics and Computer Science, Ivan Franko National University of Lviv, 79000 Lviv, Ukraine 2 School of Mathematics, University of Birmingham, Edgbaston, Birmingham B15 2TT, United Kingdom

Received  December 2007 Revised  June 2008 Published  July 2008

We consider a Cauchy problem for the Laplace equation in a two-dimensional semi-infinite region with a bounded inclusion, i.e. the region is the intersection between a half-plane and the exterior of a bounded closed curve contained in the half-plane. The Cauchy data are given on the unbounded part of the boundary of the region and the aim is to construct the solution on the boundary of the inclusion. In 1989, Kozlov and Maz'ya [10] proposed an alternating iterative method for solving Cauchy problems for general strongly elliptic and formally self-adjoint systems in bounded domains. We extend their approach to our setting and in each iteration step mixed boundary value problems for the Laplace equation in the semi-infinite region are solved. Well-posedness of these mixed problems are investigated and convergence of the alternating procedure is examined. For the numerical implementation an efficient boundary integral equation method is proposed, based on the indirect variant of the boundary integral equation approach. The mixed problems are reduced to integral equations over the (bounded) boundary of the inclusion. Numerical examples are included showing the feasibility of the proposed method.
Citation: Roman Chapko, B. Tomas Johansson. An alternating boundary integral based method for a Cauchy problem for the Laplace equation in semi-infinite regions. Inverse Problems & Imaging, 2008, 2 (3) : 317-333. doi: 10.3934/ipi.2008.2.317
References:
 [1] G. Bastay, T. Johansson, V. A. Kozlov and D. Lesnic, An alternating method for the stationary Stokes system,, ZAMM, 86 (2006), 268. doi: 10.1002/zamm.200410238. [2] J. Baumeister and A. Leitāo, On iterative methods for solving ill-posed problems modeled by partial differential equations,, J. Inv. Ill-Posed Probl., 9 (2001), 13. [3] A.-P. Calderón, Uniqueness in the Cauchy problem for partial differential equations,, Amer. J. Math., 80 (1958), 16. doi: 10.2307/2372819. [4] T. Carleman, Sur un probléme d'unicité pur les systémes d'équations aux dérivées partielles á deux variables indépendantes, (French), Ark. Mat., 26 (1939), 1. [5] R. Chapko and R. Kress, On a quadrature method for a logarithmic integral equation of the first kind, in "World Scientific Series in Applicable Analysis, Contributions in Numerical Mathematics,, Vol. 2'' (ed. Agarwal), 2 (1993), 127. [6] H. W. Engl and A. Leitāo, A Mann iterative regularization method for elliptic Cauchy problems,, Numer. Funct. Anal. Optim., 22 (2001), 861. doi: 10.1081/NFA-100108313. [7] U. Hämarik and T. Raus, On the choice of the regularization parameter in ill-posed problems with approximately given noise level of data,, J. Inverse Ill-Posed Probl., 14 (2006), 251. doi: 10.1515/156939406777340928. [8] M. A. Jawson and G. Symm, "Integral Equations Methods in Potential Theory and Elastostatics,'', Academic Press, (1977). [9] M. Jourhmane and A. Nachaoui, An alternating method for an inverse Cauchy problem,, Numer. Algorithms, 21 (1999), 247. doi: 10.1023/A:1019134102565. [10] V. A. Kozlov and V. G. Maz'ya, On iterative procedures for solving ill-posed boundary value problems that preserve differential equations,, Algebra i Analiz, 1 (1989), 144. [11] V. A. Kozlov, V. G. Maz'ya and A. V. Fomin, An iterative method for solving the Cauchy problem for elliptic equations,, Zh. Vychisl. Mat. i Mat. Fiz., 31 (1991), 64. [12] R. Kress, "Linear Integral Equations,", 2nd edition, (1999). [13] D. Lesnic, L. Elliot and D. B. Ingham, An iterative boundary element method for solving numerically the Cauchy problem for the Laplace equation,, Eng. Anal. Bound. Elem., 20 (1997), 123. doi: 10.1016/S0955-7997(97)00056-8. [14] W. McLean, "Strongly Elliptic Systems and Boundary Integral Equations,'', Cambridge University Press, (2000). [15] D. Maxwell, M. Truffer, S. Avdonin and M. Stuefer, Determining glacier velocities and stresses with inverse methods: an iterative scheme,, to appear in Journal of Glaciology., (). [16] C. Miranda, "Partial Differential Equations of Elliptic Type,'', Springer-Verlag, (1970). [17] A. Polyanin, "Handbook of Linear Partial Differential Equations for Engineers and Scientists,'', Chapman & Hall/CRC Press, (2002). [18] F. Stenger, "Numerical Methods Based on Sinc and Analytic Functions,'', Springer-Verlag, (1993). [19] G. M. Vainikko and A. Y. Veretennikov, "Iteration Procedures in Ill-Posed Problems,'', Nauka Publ., (1986).

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References:
 [1] G. Bastay, T. Johansson, V. A. Kozlov and D. Lesnic, An alternating method for the stationary Stokes system,, ZAMM, 86 (2006), 268. doi: 10.1002/zamm.200410238. [2] J. Baumeister and A. Leitāo, On iterative methods for solving ill-posed problems modeled by partial differential equations,, J. Inv. Ill-Posed Probl., 9 (2001), 13. [3] A.-P. Calderón, Uniqueness in the Cauchy problem for partial differential equations,, Amer. J. Math., 80 (1958), 16. doi: 10.2307/2372819. [4] T. Carleman, Sur un probléme d'unicité pur les systémes d'équations aux dérivées partielles á deux variables indépendantes, (French), Ark. Mat., 26 (1939), 1. [5] R. Chapko and R. Kress, On a quadrature method for a logarithmic integral equation of the first kind, in "World Scientific Series in Applicable Analysis, Contributions in Numerical Mathematics,, Vol. 2'' (ed. Agarwal), 2 (1993), 127. [6] H. W. Engl and A. Leitāo, A Mann iterative regularization method for elliptic Cauchy problems,, Numer. Funct. Anal. Optim., 22 (2001), 861. doi: 10.1081/NFA-100108313. [7] U. Hämarik and T. Raus, On the choice of the regularization parameter in ill-posed problems with approximately given noise level of data,, J. Inverse Ill-Posed Probl., 14 (2006), 251. doi: 10.1515/156939406777340928. [8] M. A. Jawson and G. Symm, "Integral Equations Methods in Potential Theory and Elastostatics,'', Academic Press, (1977). [9] M. Jourhmane and A. Nachaoui, An alternating method for an inverse Cauchy problem,, Numer. Algorithms, 21 (1999), 247. doi: 10.1023/A:1019134102565. [10] V. A. Kozlov and V. G. Maz'ya, On iterative procedures for solving ill-posed boundary value problems that preserve differential equations,, Algebra i Analiz, 1 (1989), 144. [11] V. A. Kozlov, V. G. Maz'ya and A. V. Fomin, An iterative method for solving the Cauchy problem for elliptic equations,, Zh. Vychisl. Mat. i Mat. Fiz., 31 (1991), 64. [12] R. Kress, "Linear Integral Equations,", 2nd edition, (1999). [13] D. Lesnic, L. Elliot and D. B. Ingham, An iterative boundary element method for solving numerically the Cauchy problem for the Laplace equation,, Eng. Anal. Bound. Elem., 20 (1997), 123. doi: 10.1016/S0955-7997(97)00056-8. [14] W. McLean, "Strongly Elliptic Systems and Boundary Integral Equations,'', Cambridge University Press, (2000). [15] D. Maxwell, M. Truffer, S. Avdonin and M. Stuefer, Determining glacier velocities and stresses with inverse methods: an iterative scheme,, to appear in Journal of Glaciology., (). [16] C. Miranda, "Partial Differential Equations of Elliptic Type,'', Springer-Verlag, (1970). [17] A. Polyanin, "Handbook of Linear Partial Differential Equations for Engineers and Scientists,'', Chapman & Hall/CRC Press, (2002). [18] F. Stenger, "Numerical Methods Based on Sinc and Analytic Functions,'', Springer-Verlag, (1993). [19] G. M. Vainikko and A. Y. Veretennikov, "Iteration Procedures in Ill-Posed Problems,'', Nauka Publ., (1986).
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