Numerical comparison of the LOOCV-MFS and the MS-CTM for 2D equations. (English) Zbl 1488.65716
Summary: The method of fundamental solutions (MFS) and the Collocation Trefftz method have been known as two highly effective boundary-type methods for solving homogeneous equations. Despite many attractive features of these two methods, they also experience different aspects of difficulties. Recent advances in the selection of source location of the MFS and the techniques in reducing the condition number of the Trefftz method have made significant improvement in the performance of these two methods which have been proven to be theoretically equivalent. In this paper we will compare the numerical performance of these two methods under various smoothness of the boundary and boundary conditions.
MSC:
| 65N80 | Fundamental solutions, Green’s function methods, etc. for boundary value problems involving PDEs |
| 65N35 | Spectral, collocation and related methods for boundary value problems involving PDEs |
| 65F35 | Numerical computation of matrix norms, conditioning, scaling |
Keywords:
Trefftz method; the method of fundamental solution; LOOCV; multiple scale method; non-harmonic boundary conditionsReferences:
| [1] | C. S. CHEN, A. KARAGEORGHIS, AND Y. LI, On choosing the location of the sources in the MFS, Numer. Algorithms, (2015), pp. 1-24. |
| [2] | J. T. CHEN, C. S. WU, Y. T. LEE, AND K. H. CHEN, On the equivalence of the Trefftz method and method of fundamental solutions for Laplace and biharmonic equations, Comput. Math. Appl., 53 (2007), pp. 851-879. · Zbl 1125.65110 |
| [3] | G. FAIRWEATHER AND A. KARAGEORGHIS, The method of fundamental solutions for elliptic boundary value problems, Adv. Comput. Math., 9(1-2) (1998), pp. 69-95. · Zbl 0922.65074 |
| [4] | Z. J. FU, W. CHEN, AND H. T. YANG, Boundary particle method for Laplace transformed time fractional diffusion equations, J. Comput. Phys., (235) (2013), pp. 52-66. · Zbl 1291.76256 |
| [5] | Z. J. FU AND Q. H. QIN, Three boundary meshless methods for heat conduction analysis in nonlin-ear FGMs with Kirchhoff and Laplace transformation, Adv. Appl. Math. Mech., 4(5) (2012), pp. 519-542. · Zbl 1262.65137 |
| [6] | M. A. GOLBERG AND C. S. CHEN, The method of fundamental solutions for potential, Helmholtz and diffusion problems, in M. A. Golberg, editor, Boundary Integral Methods: Numerical and Mathematical Aspects, WIT Press, 1998, pp. 103-176. · Zbl 0945.65130 |
| [7] | P. GORZELAŃCZYK AND J. A. KOŁODZIEJ, Some remarks concerning the shape of the source contour with application of the method of fundamental solutions to elastic torsion of prismatic rods, Eng. Anal. Boundary Elem., 32(1) (2008), pp. 64-75. · Zbl 1272.74623 |
| [8] | J. A. KOLODZIEJ AND A. P. ZIELINSKI, Boundary collocation method applied to non-linear condi-tions, in Boundary Collocation Techniques and Their Application in Engineering, WIT Press, 2009. |
| [9] | V. D. KUPRADZE AND M. A. ALEKSIDZE, The method of functional equations for the approximate solution of certain boundary value problems, Comput. Math. Math. Phys., 4 (1964), pp. 82-126. · Zbl 0154.17604 |
| [10] | Z. C. LI, T. T. LU, H. Y. HU, AND H. D. CHENG, The collocation Trefftz method, in Trefftz and Collocation Methods, WIT Press, 2008. · Zbl 1140.65005 |
| [11] | M. LI, C. S. CHEN, AND A. KARAGEORGHIS, The MFS for the solution of harmonic boundary value problems with non-harmonic boundary conditions, Comput. Math. Appl., 66 (2013), pp. 2400-2424. · Zbl 1350.65136 |
| [12] | Z. C. LI, T. T. LU, H. T. HUANG, AND A. H. D. CHENG, Trefftz, collocation, and other boundary methods-a comparison, Numer. Math. Partial Diff. Equations, 23(1) (2007), pp. 93-144. · Zbl 1223.65093 |
| [13] | C. S. LIU, An effectively modified direct Trefftz method for 2D potential problems considering the domain’s characteristic length, Eng. Anal. Boundary Elem., 31(12) (2007), pp. 983-993. · Zbl 1259.65183 |
| [14] | C. S. LIU, A modified Trefftz method for two-dimensional Laplace equation considering the domain’s characteristic length, CMES Comput. Model. Eng. Sci., 21(1) (2007), pp. 53-65. · Zbl 1232.65157 |
| [15] | C. S. LIU, A highly accurate MCTM for direct and inverse problems of biharmonic equation in arbitrary plane domains, CMES Comput. Model. Eng. Sci., 30(2) (2008), pp. 65-75. |
| [16] | C. S. LIU, Improving the ill-conditioning of the method of fundamental solution for 2d Laplace equation, CMES Comput. Model. Eng. Sci., 28 (2008), pp. 77-93. · Zbl 1232.65170 |
| [17] | C. S. LIU, A modified collocation Trefftz method for the inverse Cauchy problem of Laplace equation, Eng. Anal. Boundary Elem., 32(9) (2008), pp. 778-785. · Zbl 1244.65188 |
| [18] | C. S. LIU, An equilibrated method of fundamental solutions to choose the best source points for the Laplace equation, Eng. Anal. Boundary Elem., 36(8) (2012), pp. 1235-1245. · Zbl 1352.65637 |
| [19] | C. S. LIU, A multiple-scale Trefftz method for an incomplete Cauchy problem of biharmonic equation, Eng. Anal. Boundary Elem., 37(11) (2013), pp. 1445-1456. · Zbl 1287.65103 |
| [20] | C. S. LIU AND S. N. ATLURI, Numerical solution of the Laplacian Cauchy problem by using a better postconditioning collocation Trefftz method, Eng. Anal. Boundary Elem., 37(1) (2013), pp. 74-83. · Zbl 1352.65569 |
| [21] | R. MATHON AND R. L. JOHNSTON, The approximate solution of elliptic boundary-value problems by fundamental solutions, SIAM J. Numer. Anal., 14(14) (1977), pp. 638-650. · Zbl 0368.65058 |
| [22] | M. H. PROTTER AND H. F. WEINBERGER, Maximum principles in differential equations, Englewood Cliffs, 2 (1967), pp. 177-179. · Zbl 0153.13602 |
| [23] | S. RIPPA, An algorithm for selecting a good value for the parameter c in radial basis function inter-polation, Adv. Comput. Math., 11(2) (1999), pp. 193-210. · Zbl 0943.65017 |
| [24] | R. SCHABACK, Adaptive numerical solution of MFS systems, in The Method of Fundamental Solutions-A Meshless Method, volume 17, pages 1-27, Dynamic Publishers, 2007. |
| [25] | T. SHIGETA, D. L. YOUNG, AND C. S. LIU, Adaptive multilayer method of fundamental solutions using a weighted greedy QR decomposition for the Laplace equation, J. Comput. Phys., 231(21) (2012), pp. 7118-7132. |
| [26] | E. KITA AND N. KAMIYA, Trefftz method: an overview, Adv. Eng. Software, 24 (1995), pp. 3-12. · Zbl 0984.65502 |
| [27] | E. TREFFTZ, Ein gegenstück zum ritzschen verfahren, Proc. 2nd Int. Cong. Appl. Mech., Zurich, (1926), pp. 131-137. · JFM 52.0483.02 |
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