Boundary particle method: Difference between revisions

Revision as of 06:06, 10 April 2012

Revision as of 06:08, 10 April 2012

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The Boundary Particle Method (BPM) is a truly boundary-only meshless (meshfree) collocation technique, in the sense that none of inner nodes are required at all in the numerical solution of nonhomogeneous partial differential equations. Numerical experiments also show that the BPM has spectral convergence. Its interpolation matrix can be symmetric and the method is easy-to-implement and free of integration and mesh. Thanks to its boundary-only merit, the BPM has clear edge over the other numerical schemes in the solution of optimization and inverse problems, where only a part of boundary data is usually accessible.

== History and recent developments ==

In recent decades, the dual reciprocity method (DRM)[1] and multiple reciprocity method (MRM)[2] have been emerging as the two most promising techniques to evaluate the particular solution of nonhomogeneous partial differential equations in conjunction with the boundary discretization techniques, such as boundary element method (BEM). For instance, the so-called DR-BEM and MR-BEM are popular BEM techniques in the numerical solution of nonhomogeneous problems.

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Thanks to its truly boundary-only merit, the BPM is more appealing in the solution of optimization and inverse problems, where only a part of boundary data is usually accessible. The method has successfully been applied to inverse Cauchy problem associated with Poisson[9] and nonhomogeneous Helmholtz equations[10].

== Further comments ==

The BPM may encounter difficulty in the solution of problems having complex source functions, such as non-smooth, large-gradient functions, or a set of discrete measured data. The road map for the BPM solution of such problems is briefly outlined below:

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(2) The ___domain decomposition may be used to in the BPM boundary-only solution of large-gradient source functions problems.

== See also ==

*  Meshfree method