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Keywords
(12)
Approximate Solution
Boundary Condition
Boundary Value Problem
Collocation Method
Equivalence Relation
Fundamental Solution
Laplace Equation
Linear Equations
Method of Fundamental Solution
Numerical Method
Numerical Solution
trefftz method
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(1)
Some comments on the illconditioning of the method of fundamental solutions
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Improving the Illconditioning of the Method of Fundamental Solutions for 2D Laplace Equation
Improving the Illconditioning of the Method of Fundamental Solutions for 2D Laplace Equation,CheinShan Liu
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Improving the Illconditioning of the Method of Fundamental Solutions for 2D Laplace Equation
(
Citations: 9
)
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CheinShan Liu
The method of fundamental solu tions (MFS) is a truly meshless
numerical method
widely used in the elliptic type boundary value problems, of which the
approximate solution
is expressed as a linear combination of fundamental solutions and the unknown coefficients are deter mined from the boundary conditions by solving a
linear equations
system. However, the accuracy of MFS is severely limited by its illconditioning of the resulting
linear equations
system. This paper is motivated by the works of Chen, Wu, Lee and Chen (2007) and Liu (2007a). The first paper proved an equivalent relation of the Tre fftz method and MFS for circular domain, while the second proposed a modified
Trefftz method
(MTM). We first prove an equivalent relation of MTM and MFS for arbitrary plane domain. Due to the wellposedness of MTM, we can alleviate the illconditioning of MFS through a new
linear equations
system of the modified MFS (MMFS). In doing so we can raise the accuracy of MMFS over four orders more than the original MFS. Nu merical examples indicate that the MMFS can attain highly accurate numerical solutions with accuracy over the order of 10−10. The present method is fully not similar to the preconditioning technique as used to solve the illconditioned lin ear equations system.
Published in 2009.
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Citation Context
(5)
...(
2004
), Liu and Gu (
2005
), Liu (
2008
), Nguyen
et al
...
Daohong Ding
,
et al.
Combination of Newmark method and natural element method for elastodyn...
...Previously, Liu
50
has proposed a new preconditioner to reduce the illcondition of the MFS...
CheinShan Liu
.
An optimal trivector iterative algorithm for solving illposed linear...
...Then, Liu [
37
] found that the proper property of the modified Trefftz method can be used to modify the method of fundamental solutions (MFS) by relating the coefficients obtained from these two methods by a linear transformation, as shown in Eq...
CheinShan Liu
.
The Method of Fundamental Solutions for Solving the Backward Heat Cond...
...To overcome mesh generation problem, facing the traditional meshbased BEM, a variety of boundarytype meshless methods, in recent years, have been developed and attract a lot of attention, for example, method of fundamental solution (MFS) [4, 11,
22
], boundary knot method (BKM) [10], plane wave method (PWM) [3, 29], collocation Trefftz method (CTM) [21] and regularized meshless method (RMM) [5, 32], among which the former three methods ...
Wen Chen
,
et al.
BOUNDARY PARTICLE METHOD FOR INVERSE CAUCHY PROBLEMS OF INHOMOGENEOUS ...
...
Liu (2008)
has proved an equivalent relation between the modified Trefftz method (MTM) and MFS for arbitrary plane domains and, in addition, has shown that the illconditioning of the MFS can be alleviated through the MTM by obtaining a new system of linear equations for the corresponding modified MFS...
Liviu Marin
.
Stable MFS Solution to Singular Direct and Inverse Problems Associated...
References
(34)
Fundamental Solutions Method for Elliptic Boundary Value Problems
(
Citations: 145
)
Alexander Bogomolny
Journal:
Siam Journal on Numerical Analysis  SIAM J NUMER ANAL
, vol. 22, no. 4, pp. 644669, 1985
A Modified Trefftz Method for TwoDimensional Laplace Equation Considering the Domain's Characteristic Length
(
Citations: 15
)
CheinShan Liu
Published in 2007.
A Meshless Regularized Integral Equation Method (MRIEM) for Laplace Equation in Arbitrary Interior or Exterior Plane Domains
(
Citations: 10
)
CheinShan Liu
Published in 2007.
A highly accurate collocation Trefftz method for solving the Laplace equation in the doubly connected domains
(
Citations: 11
)
CheinShan Liu
Journal:
Numerical Methods for Partial Differential Equations  NUMER METHOD PARTIAL DIFFER E
, vol. 24, no. 1, pp. 179192, 2008
A critical assessment of the truly Meshless Local PetrovGalerkin (MLPG), and Local Boundary Integral Equation (LBIE) methods
(
Citations: 138
)
S. N. Atluri
,
H.G. Kim
,
J. Y. Cho
Journal:
Computational Mechanics  COMPUTATION MECH
, vol. 24, no. 5, pp. 348372, 1999
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Citations
(9)
Combination of Newmark method and natural element method for elastodynamics
Daohong Ding
,
Qing Zhang
,
Fuzhang Wang
,
Lingyan Luo
Journal:
Journal of The Chinese Institute of Engineers  J CHIN INST ENGINEERS
, vol. aheadofp, no. aheadofp, pp. 19, 2012
An optimal trivector iterative algorithm for solving illposed linear inverse problems
CheinShan Liu
Journal:
Inverse Problems in Science and Engineering  INVERSE PROBL SCI ENG
, vol. aheadofp, no. aheadofp, pp. 132, 2012
The Method of Fundamental Solutions for Solving the Backward Heat Conduction Problem with Conditioning by a New PostConditioner
(
Citations: 2
)
CheinShan Liu
Journal:
Numerical Heat Transfer Part Bfundamentals  NUMER HEAT TRANSFER PT BFUND
, vol. 60, no. 1, pp. 5772, 2011
Mathematical and Numerical Studies on Meshless Methods for Exterior Unbounded Domain Problems
Takemi Shigeta
,
D. L. Young
Published in 2010.
BOUNDARY PARTICLE METHOD FOR INVERSE CAUCHY PROBLEMS OF INHOMOGENEOUS HELMHOLTZ EQUATIONS
(
Citations: 5
)
Wen Chen
,
Zhuojia Fu
Published in 2009.