Interdisciplinary Applied Mathematics

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In summary, domain meshless methods are attractive alternatives to traditional numerical methods. The implementation of boundary conditions in Galerkin-based meshless methods has some difficulties, but a number of approaches have been suggested to overcome these difficulties (see, e.g., (Li and Liu, 2002), and references therein). The collocation meshless methods can impose the boundary conditions exactly, but the robustness of the method can be an issue for random distribution of points. This issue has been addressed in (Jin et al., 2004), but more progress is desirable. The construction of approximation functions is more expensive in meshless methods compared to the cost associated with construction of interpolation functions in the finite element method. The integration cost in Galerkin meshless methods is more expensive. Galerkin meshless methods can be a few times slower (typically about 5 times) than finite element methods. Collocation meshless methods are much faster, since no numerical integrations are involved. However, they may need more points, and as mentioned above, the robustness needs to be improved. For a quadratic basis in 2-D, the collocation meshless method has been shown to converge quadratically. Even though not much work has been published on time-stepping schemes in meshless methods, much of the published literature on time-stepping schemes for finite difference and finite element methods is applicable to meshless methods.

14-2.2 Boundary-Only Simulation

The key idea in boundary-only simulation is to discretize only the boundary of the problem. For linear and exterior problems (infinite domain problems), where there is a well-defined Green’s function, boundary integral formulations are attractive, since they need only the discretization of a surface. The discretization of a surface into a mesh can, however, be quite involved for

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