Meshfree methods are generally formulated under two frameworks: the popular Galerkin weak form, and strong form collocation \[1\]. The latter approach leverages the flexible properties of meshfree approximations, which can obtain arbitrary-order smoothness effortlessly. In the Galerkin weak-form approaches, several issues arise. Essential boundary conditions are non-trivial, since in general meshfree approximations do not possess the Kronecker delta property, or even the weak Kronecker delta property \[2\]. Thus considerable effort has gone into researching the issue of boundary conditions. In addition, since the weak form is utilized to formulate the problem, quadrature becomes an issue \[3,4\]. If traditional Gauss quadrature is employed, then high-order rules are needed to ensure convergence. On the other hand, to keep the meshfree character of the method, nodal integration is often desired, but is of course a low-order quadrature, which requires both stabilization and correction for integration accuracy. Thus several methods have been developed in order to yield an efficient, stable, and accurate Galerkin solution.

Collocation methods on the other hand, do not require these special treatments; essential boundary conditions are trivial, and quadrature is not an issue. However, since they are based on the strong form, higher-order derivatives are required to formulate the solution, which are computationally expensive \[5\]. In addition, many collocation points are needed in order to ensure optimal convergence rates, analogous for the necessity of high-order quadrature in weak-form based methods, and this also increases the cost of the method.

Since both methods have advantageous and disadvantageous in terms of accuracy and computational complexity, this begs the question of which method is a more effective approach. That is, for a given level of error, which approach requires the least amount of computational time, or vice versa. In this work, complexity analyses of both methods is presented. These analyses include various state-of-the-art corrections and stabilizations needed to ensure an effective solution is obtained. Operation counts are given and compared to one another, and the level of error is assessed for each method numerically. Several comparisons are made, and solution strategies are suggested, such as which method is the most effective when low computational time is desired, or when the lowest possible error is desired.

\[1\] Chen, Jiun-Shyan, Michael Hillman, and Sheng-Wei Chi. "Meshfree methods: progress made after 20 years." Journal of Engineering Mechanics 143.4 (2017): 04017001.

\[2\] Fernández-Méndez, Sonia, and Antonio Huerta. "Imposing essential boundary conditions in mesh-free methods." Computer methods in applied mechanics and engineering 193.12-14 (2004): 1257-1275.

\[3\] Chen, J. S., Wu, C. T., Yoon, S., & You, Y. (2001). A stabilized conforming nodal integration for Galerkin mesh‐free methods. *International journal for numerical methods in engineering*, *50*(2), 435-466.

\[4\] Hillman, M., & Chen, J. S. (2016). An accelerated, convergent, and stable nodal integration in Galerkin meshfree methods for linear and nonlinear mechanics. *International Journal for Numerical Methods in Engineering*, *107*(7), 603-630.

\[5\] Hu, Hsin-Yun, Chiu-Kai Lai, and Jiun-Shyan Chen. "A study on convergence and complexity of reproducing kernel collocation method." National Science Council Tunghai University Endowment Fund for Academic Advancement Mathematics Research Promotion Center (2009): 189.


Complexity Analyses of State-of-the-Art Galerkin and Collocation Meshfree Methods

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