Computational Mechanics

, Volume 64, Issue 5, pp 1421–1454 | Cite as

A gradient reproducing kernel collocation method for high order differential equations

  • Ashkan Mahdavi
  • Sheng-Wei ChiEmail author
  • Huiqing Zhu
Original Paper


The High order Gradient Reproducing Kernel in conjunction with the Collocation Method (HGRKCM) is introduced for solutions of 2nd- and 4th-order PDEs. All the derivative approximations appearing in PDEs are constructed using the gradient reproducing kernels. Consequently, the computational cost for construction of derivative approximations reduces tremendously, basis functions for derivative approximations are smooth, and the accumulated error arising from calculating derivative approximations are controlled in comparison to the direct derivative counterparts. Furthermore, it is theoretically estimated and numerically tested that the same number of collocation points as the source points can be used to obtain the optimal solution in the HGRKCM. Overall, the HGRKCM is roughly 10–25 times faster than the conventional reproducing kernel collocation method. The convergence of the present method is estimated using the least squares functional equivalence. Numerical results are verified and compared with other strong-form-based and Galerkin-based methods.


Strong form collocation Weighted collocation method Gradient reproducing kernel Reproducing kernel collocation method 



Research reported in this paper was partially supported by DoD SERDP under contract number W912HQ18C0099.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Civil and Material EngineeringUniversity of Illinois at ChicagoChicagoUSA
  2. 2.Department of MathematicsUniversity of Southern MississippiHattiesburgUSA

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