In this paper, the effect of strong discontinuities on the solution of a Dirichlet problem governed by the Poisson equation is investigated. The numerical solution is obtained using the local differential quadrature method (LDQM), in two ways: one employs the conventional LDQM approach without any discontinuity treatment, and the second uses a new LDQM technique to treat the discontinuity, which is based on reducing the propagation of errors arising from the discontinuity. The results show that the LDQM solution achieves greater accuracy when the discontinuity treatment technique is used.
ShuCDingHYeoS. Local radial basis function-based differential quadrature method and its application to solve two-dimensional incompressible Navier–Stokes equations. Comput Methods Appl Mech Eng2003; 192(7–8): 941–954.
4.
TolstykhAShirobokovDA. On using radial basis functions in a “finite difference mode” with applications to elasticity problems. Comput Mech2003; 33: 68–79.
5.
DingHShuCYeoKS, et al. Numerical computation of three-dimensional incompressible viscous flows in the primitive variable form by local multiquadric differential quadrature method. Comput Methods Appl Mech Eng2006; 195: 516–533.
6.
ShuCShanYQinN. Development of a local MQ-DQ-based stencil adaptive method and its application to solve incompressible Navier–Stokes equations. Int J Numer Methods Fluids2007; 55: 367–386.
7.
ShanYYShuCLuZL. Application of local MQ-DQ method to solve 3D incompressible viscous flows with curved boundary. Comput Model Eng Sci2008; 25: 99–99.
8.
ZhuHShuHDingM. Numerical solutions of two-dimensional Burgers’ equations by discrete Adomian decomposition method. Comput Math Appl2010; 60: 840–848.
9.
JaśkowiecJMilewskiS. Coupling finite element method with meshless finite difference method by means of approximation constraints. Comput Math Appl2023; 142: 208–224.
10.
AhmadIZamanSShakeelM, et al. Local meshless differential quadrature collocation method for time-fractional PDEs. Discrete Contin Dyn Syst–S2020; 13: 2641–2654.
11.
DingHShuCTangDB. Error estimates of local multiquadric-based differential quadrature (LMQDQ) method through numerical experiments. Int J Numer Methods Eng2005; 63: 1513–1529.
12.
SilvaJRSantosLGCManzanaresN. A local differential quadrature method with variable shape multiquadrics: tests on Poisson equation and fluid dynamics using consistent cloud refinements. RETERM2017; 16: 62–66.
13.
SilvaJRFariaJGPAfonsoMM, et al. Effect of local support configuration on the precision of numerical solutions of Poisson equation obtained with differential quadrature method. IEEE Trans Magn2017; 53: 1–1.
14.
SilvaJRAfonsoMMFariaJGP. Numerical treatment of electrostatic contour value problem in heterogeneous media using local differential quadrature method. In: 19th international symposium on electromagnetic fields in mechatronics, electrical and electronic engineering (ISEF), Nancy, France, 29–31 August 2019.
15.
FrankeR. Scattered data interpolation: tests of some methods. Math Comput1982; 38: 181–199.
16.
FornbergBWrightG. Stable computation of multiquadric interpolants for all values of the shape parameter. Comput Math Appl2004; 48(5–6).
17.
SarraSASturgillD. A random variable shape parameter strategy for radial basis function approximation methods. Eng Anal Bound Elem2009; 33: 1239–1245.
18.
OliverJ. Modelling strong discontinuities in solid mechanics via strain softening constitutive equations. Part 1: fundamentals. Int J Num Methods Eng1996; 39: 3575–3600.
19.
AltenbachHÖchsnerA (eds). Encyclopedia of continuum mechanics. Cham: Springer, 2020.
20.
LenarduzziLSchabackR. Kernel-based adaptive approximation of functions with discontinuities. Appl Math Comput2017; 307: 113–123.
21.
CecilTC. Numerical methods for partial differential equations involving discontinuities. PhD Thesis, University of California, Los Angeles, CA, 2003.
22.
DehghanMNikpourA. Numerical solution of the system of second-order boundary value problems using the local radial basis functions based differential quadrature collocation method. Appl Math Modell2013; 37(18–19).
23.
BayonaVMoscosoMCarreteroM, et al. RBF-FD formulas and convergence properties. J Comput Phys2010; 229: 8281–8295.
24.
FungTC. Stability and accuracy of differential quadrature method in solving dynamic problems. Comput Methods Appl Mech Eng2002; 191: 1311–1331.
25.
NgCHWZhaoYBXiangY, et al. On the accuracy and stability of a variety of differential quadrature formulations for the vibration analysis of beams. Int J Eng Appl Sci2009; 1: 1–25.
26.
FornbergBWrightGLarssonE. Some observations regarding interpolants in the limit of flat radial basis functions. Comput Math Appl2004; 47: 37–55.
27.
AragónAMSimoneA. The discontinuity-enriched finite element method. Int J Numer Methods Eng2017; 112: 1589–1613.