A Regularized Trefftz Method for Solving an Inverse Problem for the Laplace Equation

Abdelhak Hadj*; Hacene Saker

**Abdelhak Hadj* and Hacene Saker**

Volume3-Issue5
Dates: Received: 2022-05-04 | Accepted: 2022-05-10 | Published: 2022-05-13
Pages: 522-536

Abstract

The present paper proposes a new regularised Trefftz method to recover the boundary value on a non-accessible boundary of an annulus from overdetermined data on the accessible boundary of that annulus. Considering that the available data have a Fourier expansion, we consider the Tikhonov damping factor in constructing our regularized scheme, which satisfies the boundary value problem. At the same time, the convergence estimates for the regularised solution will be established under an assumption for the exact solution. The finite term truncation of the series expansion allows us to match the boundary condition as accurately as desired. By the collocation method, we derive a system of linear equations that can be uniquely solved to obtain the coefficients and preset the regularisation parameter and the damping factor to ensure adequate stability. The numerical efficiency of the proposed method is investigated with a high truncation number in comparison with the modified collocation Trefftz method.

FullText HTML FullText PDF DOI: 10.37871/jbres1475

Certificate of Publication

Copyright

How to cite this article

Hadj A, Saker H. A Regularized Trefftz Method for Solving an Inverse Problem for the Laplace Equation. J Biomed Res Environ Sci. 2022 May 13; 3(5): 522-536. doi: 10.37871/jbres1475, Article ID: JBRES1475, Available at: https://www.jelsciences.com/articles/jbres1475.pdf

Subject area(s)

Computer Science

References

Liu CS. A highly accurate collocation Trefftz method for solving the Laplace equation in the doubly connected domains. Numeric Meth Part diff Eq. 2008;24(1):179-192. doi: 10.1002/num.20257.
Ramm AG. A geometrical inverse problem. Inv Prob. 1986;2(2):19.
Karageorghis A, Lesnic D, Marin L. Regularized collocation Trefftz method for void detection in two-dimensional steady-state heat conduction problems. In Pres in Sc and Engine. 2014;22(3):95-418.
Liu CS. A highly accurate MCTM for direct and inverse problems of biharmonic equation in arbitrary plane domains. CMES. 2008;30(2):65-75. https://bit.ly/3MbKoFq
Kirsch A. An Introduction to the mathematical theory of inverse problems. Springer. 2011.
Ott KA, Brown RM. The mixed problem for the Laplacian in Lipschitz domains. Potential Anal. 2013;38:1333-1364. https://bit.ly/3McyU4E
Lanzani L, Capogna L, Brown RM. The mixed problem in L for some two-dimensional Lipschitz domains. Math Ann. 2008;342:91-124.
Lanzani L, Capogna L, Brown RM. The solution of mixed boundary value problem for the Laplace equation in a multiply connected domain. Probl Anal Issues Anal. 2019;26(2):51-66.
Kress R. Linear Integral Equations. Springer. 2014.
Hadamard J. Lecture note on Cauchys problem in linear partial differential equations. Yale University Press; 1923.
Tajani C, Abouchabaka J. On the data completion problem for Laplace’s equation. Annals of the Univ of Cra, Math and Comp Sci Ser. 2018;45(1):11-36. https://bit.ly/3wn7Rgj
Hadj A, Saker H. Integral equations method for solving a Biharmonic inverse problem in detection of Robin coefficients. Applied Numerical Mathematics. 2021;160:436-450. doi: 10.1016/j.apnum.2020.10.005
Ausaru A, Nagarani P. Effect of external body acceleration on solute dispersion in unsteady non-newtonian fluid flow-the generalized dispersion model approach. Int J Appl Comput Math. 2022;8:13. doi: 10.1007/s40819-021-01209-w.
Li ZC, Lu TT, Huang HT, Cheng AHD. Trefftz, collocation, and other boundary methods - A comparison. Numeric Meths for Part Diff Eq. 2008;23(1):93-144. doi: 10.1002/num.20159.
Chen CT, Chen KH, Lee JF, Chen JT. Adaptive error estimation of the Trefftz method for solving the Cauchy problem. WIT Press. 2007;44:14. doi: 10.2495/BE070051.
Liu CS. A modified Trefftz Method for Two-Dimensional Laplace Equation considering the domain’s characteristic length. CMES. 2007;21(1):53-65. doi: 10.3970/cmes.2007.021.053.
Liu CS. A modified collocation Trefftz method for the inverse Cauchy problem of Laplace equation. Engine anal with Bound elem. 2008;32(9):778-785.
Karageorghis A. Efficient Trefftz collocation algorithms for elliptic problems in circular domains. Numer Algor. 2013;64:427-453. doi: 10.1007/s11075-012-9673-8.
Liu CS. Solving the inverse problems of Laplace equation to determine the robin coefficient cracks position inside a disk. CMES. 2009;40(1):1-28 .
Liu CS, Wang F, Gu Y. Trefftz energy method for solving the Cauchy problem of the Laplace equation. App Math Lets. 2018;79:187-195.
Fan CM, Chan HF. Modified collocation Trefftz method for geometry boundary identification problem of heat conduction. Numerical Heat Transfer. 2011;59(1):58-75. doi: 10.1080/10407790.2010.541355.
Kita E, Kamiya N. Trefftz method: An overview. Adv Eng Software. 1995;24(1):3-12.
Yeih W, Liu CS, Kuo CL, Atluri SN. On solving the direct/inverse cauchy problems of Laplace equation in a multiply connected domain, using the generalized multiple-source-point boundary-collocation Trefftz Method and characteristic lengths. Computers, Materials & Continua. 2010;17(3):275-302. doi: 10.3970/cmc.2010.017.275.
Thirumaran V, Weliwita JA, Ishak MIM. An Analysis of axial couette flow in annular region of abruptly stopped pipes. Phy Sci Inter Jour. 2018;17(4):1-12. doi: 10.9734/PSIJ/2018/40076.
Benrabah A, Boussetila N. Modified nonlocal boundary value problem method for an ill-posed problem for the biharmonic equation. Inv Probs in Sci and Engi. 2018;340-368. doi: 10.1080/17415977.2018.1461859.