Abstract
A powerful method of the solution of homogeneous equations is considered. Using the traditional approach of the Method of Fundamental Solutions, the fundamental solution has to be shifted to external source points. This is inconvenient from computational point of view, moreover, the resulting linear system can easily become severely ill-conditioned. To overcome this difficulty, a special regularization technique is applied. In this approach, the original second-order elliptic problem (a modified Helmholtz problem in the paper) is approximated by a fourth-order multi-elliptic boundary interpolation problem. To perform this boundary interpolation, either the Method of Fundamental Solutions, or a direct multi-elliptic interpolation can be used. In the paper, a priori error estimations are deduced. A numerical example is also presented.
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References
C.J.S. Alves, C.S. Chen, B. Sarler, The Method of Fundamental Solutions for Solving Poisson Problems, ed. by C.A. Brebbia, A. Tadeu, V. Popov. International Series on Advances in Boundary Elements, vol. 13 (WIT, Southampton, 2002), pp. 67–76
I. Babuška, A.K. Aziz, Survey Lectures on the Mathematical Foundations of the Finite Element Method, ed. by A.K. Aziz. The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations (Academic, New York, 1972)
W. Chen, L.J. Shen, Z.J. Shen, G.W. Yuan, Boundary knot method for poisson equations. Eng. Anal. Boundary Elem. 29/8, 756–760 (2005)
W. Chen, F.Z. Wang, A method of fundamental solutions without fictitious boundary. Eng. Anal. Boundary Elem. 34, 530–532 (2010)
C. Gáspár, Multi-level biharmonic and bi-Helmholtz interpolation with application to the boundary element method. Eng. Anal. Boundary Elem. 24/7–8, 559–573 (2000)
C. Gáspár, Fast Multi-level Meshless Methods Based on the Implicit Use of Radial Basis Functions, ed. by M. Griebel, M.A. Schweitzer. Lecture Notes in Computational Science and Engineering, Vol. 26 (Springer, New York, 2002), pp. 143–160.
C. Gáspár, A Multi-Level Regularized Version of the Method of Fundamental Solutions, ed. by C.S. Chen, A. Karageorghis, Y.S. Smyrlis. The Method of Fundamental Solutions—A Meshless Method (Dynamic, Atlanta, 2008), pp. 145–164
C. Gáspár, Multi-level meshless methods based on direct multi-elliptic interpolation. J. Comput. Appl. Math. 226/2, 259–267 (2009). Special issue on large scale scientific computations
T.K. Nilssen, X.-C. Tai, R. Winther, A robust nonconforming H 2-Element. Math. Comput. 70/234, 489–505 (2000)
B. Šarler, Solution of potential flow problems by the modified method of fundamental solutions: formulations with the single layer and the double layer fundamental solutions. Eng. Anal. Boundary Elem. 33, 1374–1382 (2009)
D.L. Young, K.H. Chen, C.W. Lee, Novel meshless method for solving the potential problems with arbitrary domain. J. Computat. Phys. 209, 290–321 (2005)
Acknowledgements
The research was partly supported by the European Union (co-financed by the European Social Fund) under the project TÁMOP 4.2.1/B-09/1/KMR-2010-0003.
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Gáspár, C. (2013). Some Regularized Versions of the Method of Fundamental Solutions. In: Griebel, M., Schweitzer, M. (eds) Meshfree Methods for Partial Differential Equations VI. Lecture Notes in Computational Science and Engineering, vol 89. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32979-1_12
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DOI: https://doi.org/10.1007/978-3-642-32979-1_12
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