# Boundary-Integral Approach to the Numerical Solution of the Cauchy Problem for the Laplace Equation

- 46 Downloads

We present a survey of a direct method of boundary integral equations for the numerical solution of the Cauchy problem for the Laplace equation in doubly connected domains. The domain of solution is located between two closed boundary surfaces (curves in the case of two-dimensional domains). This Cauchy problem is reduced to finding the values of a harmonic function and its normal derivative on one of the two closed parts of the boundary according to the information about these quantities on the other boundary surface. This is an ill-posed problem in which the presence of noise in the input data may completely destroy the procedure of finding the approximate solution. We describe and present the results for a procedure of regularization aimed at the stable determination of the required quantities based on the representation of the solution to the Cauchy problem in the form a single-layer potential. For given data, this representation yields a system of boundary integral equations with two unknown densities. We establish the existence and uniqueness of these densities and propose a method for the numerical discretization in two- and three-dimensional domains. We also consider the cases of simply connected domains of the solution and unbounded domains. Numerical examples are presented both for two- and three-dimensional domains. These numerical results demonstrate that the proposed method gives good accuracy with relatively small amount of computations.

## Preview

Unable to display preview. Download preview PDF.

## References

- 1.C. Babenko, R. Chapko, and B. T. Johansson, “On the numerical solution of the Laplace equation with complete and incomplete Cauchy data using integral equations,”
*Comput. Model. Eng. Sci.*,**101**, 299–317 (2014).MathSciNetMATHGoogle Scholar - 2.C. Babenko, R. Chapko, and B. T. Johansson,
*On the Numerical Solution of the Cauchy Problem for the Laplace Equation in a Toroidal Domain by a Boundary Integral Equation Method*, Bucharest, Romania: Editura Acad. (to appear).Google Scholar - 3.T. N. Baranger, B. T. Johansson, and R. Rischette, “On the alternating method for Cauchy problems and its finite-element discretization,”
*Springer Proc. Math. Stat.*, 183–197 (2013).Google Scholar - 4.I. Borachok, R. Chapko, and B. T. Johansson, “Numerical solution of an elliptic three-dimensional Cauchy problem by the alternating method and boundary integral equations,”
*J. Inverse Ill-Posed Probl.*, 144–148 (2016).Google Scholar - 5.Y. Boukari and H. Haddar, “A convergent data completion algorithm using surface integral equations,”
*Inverse Problems*,**31**, 035011 (2015).MathSciNetCrossRefMATHGoogle Scholar - 6.F. Cakoni and R. Kress, “Integral equations for inverse problems in corrosion detection from partial Cauchy data,”
*Inverse Probl. Imaging*,**1**, 229–245 (2007).MathSciNetCrossRefMATHGoogle Scholar - 7.H. Cao, M. V. Klibanov, and S. V. Pereverzev, “A Carleman estimate and the balancing principle in the quasi-reversibility method for solving the Cauchy problem for the Laplace equation,”
*Inverse Problems*,**25**, 1–21 (2009).MathSciNetCrossRefMATHGoogle Scholar - 8.R. Chapko and B. T. Johansson, “An alternating boundary integral based method for a Cauchy problem for the Laplace equation in semi-infinite domains,”
*Inverse Probl. Imaging*,**2**, 317–333 (2008).MathSciNetCrossRefMATHGoogle Scholar - 9.R. Chapko and B. T. Johansson, “On the numerical solution of a Cauchy problem for the Laplace equation via a direct integral equation approach,”
*Inverse Probl. Imaging*,**6**, 25–36 (2012).MathSciNetCrossRefMATHGoogle Scholar - 10.R. Chapko and B. T. Johansson, “A direct integral equation method for a Cauchy problem for the Laplace equation in threedimensional semi-infinite domains,”
*Comput. Model. Eng. Sci.*,**85**, 105–128 (2012).MATHGoogle Scholar - 11.R. Chapko, B. T. Johansson, and Y. Savka, “Integral equation method for the numerical solution of the Cauchy problem for the Laplace equation in a doubly connected planar domain,”
*Inverse Probl. Sci. Eng.*,**22**, 130–149 (2014).MathSciNetCrossRefMATHGoogle Scholar - 12.M. Costabel, “Some historical remarks on the positivity of boundary integral operators,” in: M. Schanz and O. Steinback (Eds.),
*Boundary Element Analysis: Lect. Notes Appl. Comput. Mech.*, 29, Springer, Berlin (2007), pp. 1–27.CrossRefGoogle Scholar - 13.D.-N. Hào, B. T. Johansson, D. Lesnic, and P.-M. Hien, “A variational method and approximations of a Cauchy problem for elliptic equations,”
*J. Algorithms Comput. Technol.*,**4**, 89–119 (2010).MathSciNetCrossRefMATHGoogle Scholar - 14.P. C. Hansen,
*Rank-Deficient and Discrete Ill-Posed Problems. Numerical Aspects of Linear Inversion*, SIAM, Philadelphia, PA (1998).Google Scholar - 15.P. C. Hansen, “The
*L*-curve and its use in the numerical treatment of inverse problems,” in: P. Johnston (editor),*Comput. Inverse Probl. Electrocardiol*. , WIT Press, Southampton (2001), pp. 119–142.Google Scholar - 16.M. Ganesh and I. G. Graham, “A high-order algorithm for obstacle scattering in three dimensions,”
*J. Comput. Phys.*,**198**, 211–242 (2004).MathSciNetCrossRefMATHGoogle Scholar - 17.J. Hadamard, “Sur les problemes aux derivees partielles et leur signification physique,”
*Princeton Univ. Bull.*,**13**, 49–52 (1902).MathSciNetGoogle Scholar - 18.V. Isakov,
*Inverse Problems for Partial Differential Equations*, Springer, New York (1998).CrossRefMATHGoogle Scholar - 19.S. I. Kabanikhin and A. L. Karchevsky, “Optimization method for solving the Cauchy problem for an elliptic equation,”
*J. Inverse Ill-Posed Probl.*,**3**, 21–46 (1995).MathSciNetMATHGoogle Scholar - 20.S. I. Kabanikhin and M. A. Shishlenin, “Direct and iteration methods for solving inverse and ill-posed problems,”
*Sib. Èlektron. Mat. Izv.*, 595–608 (2008).Google Scholar - 21.A. Karageorghis, D. Lesnic, and L. Marin, “A survey of applications of the MFS to inverse problems,”
*Inv. Probl. Sci. Eng.*,**19**, 309–336 (2011).MathSciNetCrossRefMATHGoogle Scholar - 22.A. Kirsch,
*An Introduction to the Mathematical Theory of Inverse Problems*, Second Edition, Springer, New York (2011).CrossRefMATHGoogle Scholar - 23.V. A. Kozlov and V. G. Maz’ya, “On iterative procedures for solving ill-posed boundary-value problems that preserve differential equations,”
*Algebra Anal.*,**1**, 144–170 (1989);**English translation***: Leningrad Math. J.*,**1**, 1207–1228) (1990).Google Scholar - 24.R. Kress, “A Nystr¨om method for boundary integral equations in domains with corners,”
*Numer. Math.*,**58**, 145–161 (1990).MathSciNetCrossRefMATHGoogle Scholar - 25.R. Kress,
*Linear Integral Equations*, 3rd Edn., Springer, Heidelberg (2013).MATHGoogle Scholar - 26.D. Lesnic, L. Elliott, and D. B. Ingham, “An iterative boundary element method for solving numerically the Cauchy problem for the Laplace equation,”
*Eng. Anal. Bound. Elem.*,**20**, 123–133 (1997).CrossRefGoogle Scholar - 27.L. Marin, L. Elliott, D. B. Ingham, and D. Lesnic, “Boundary element method for the Cauchy problem in linear elasticity,”
*Eng. Anal. Bound. Elem.*,**25**, 783–793 (2001).CrossRefMATHGoogle Scholar - 28.W. McLean,
*Strongly Elliptic Systems and Boundary Integral Operators*, Cambridge Univ. Press, Cambridge (2000).MATHGoogle Scholar - 29.L. Wienert, “Die Numerische Approximation von Randintegraloperatoren für die Helmholtzgleichung im
**R**^{3}*,*” Ph. D. Thesis, Univ. Göttingen (1990).Google Scholar