TY - JOUR
T1 - Numerical solution of the Laplacian Cauchy problem by using a better postconditioning collocation Trefftz method
AU - Liu, Chein Shan
AU - Atluri, Satya N.
N1 - Funding Information:
The authors highly appreciate the constructive comments from anonymous referees, which improve the quality of this paper. The Project NSC-99-2221-E-002-074-MY3 and the 2011 Outstanding Research Award from National Science Council of Taiwan , and Taiwan Research Front Awards 2011 from Thomson Reuters granted to the first author are also highly appreciated. This work at UCI was supported by the Vehicle Army Research Labs , under a Collaborative Research Agreement with UCI .
PY - 2013/1
Y1 - 2013/1
N2 - In this paper, the inverse Cauchy problem for Laplace equation defined in an arbitrary plane domain is investigated by using the collocation Trefftz method (CTM) with a better postconditioner. We first introduce a multiple-scale Rk in the T-complete functions as a set of bases to expand the trial solution. Then, the better values of Rk are sought by using the concept of an equilibrated matrix, such that the resulting coefficient matrix of a linear system to solve the expansion coefficients is best-conditioned from a view of postconditioner. As a result, the multiple-scale Rk can be determined exactly in a closed-form in terms of the collocated points used in the collocation to satisfy the boundary conditions. We test the present method for both the direct Dirichlet problem and the inverse Cauchy problem. A significant reduction of the condition number and the effective condition number can be achieved when the present CTM is used, which has a good efficiency and stability against the disturbance from large random noise, and the computational cost is much saving. Some serious cases of the inverse Cauchy problems further reveal that the unknown data can be recovered very well, although the overspecified data are provided only at a 20% of the overall boundary.
AB - In this paper, the inverse Cauchy problem for Laplace equation defined in an arbitrary plane domain is investigated by using the collocation Trefftz method (CTM) with a better postconditioner. We first introduce a multiple-scale Rk in the T-complete functions as a set of bases to expand the trial solution. Then, the better values of Rk are sought by using the concept of an equilibrated matrix, such that the resulting coefficient matrix of a linear system to solve the expansion coefficients is best-conditioned from a view of postconditioner. As a result, the multiple-scale Rk can be determined exactly in a closed-form in terms of the collocated points used in the collocation to satisfy the boundary conditions. We test the present method for both the direct Dirichlet problem and the inverse Cauchy problem. A significant reduction of the condition number and the effective condition number can be achieved when the present CTM is used, which has a good efficiency and stability against the disturbance from large random noise, and the computational cost is much saving. Some serious cases of the inverse Cauchy problems further reveal that the unknown data can be recovered very well, although the overspecified data are provided only at a 20% of the overall boundary.
KW - Collocation Trefftz method (CTM)
KW - Ill-posed problem
KW - Inverse Cauchy problem
KW - Laplace equation
KW - Multiple-scale Trefftz method (MSTM)
KW - Postconditioner
UR - http://www.scopus.com/inward/record.url?scp=84866777004&partnerID=8YFLogxK
U2 - 10.1016/j.enganabound.2012.08.008
DO - 10.1016/j.enganabound.2012.08.008
M3 - Article
AN - SCOPUS:84866777004
VL - 37
SP - 74
EP - 83
JO - Engineering Analysis with Boundary Elements
JF - Engineering Analysis with Boundary Elements
SN - 0955-7997
IS - 1
ER -