TY - JOUR
T1 - Weighted least-squares collocation methods for elliptic PDEs with mixed boundary conditions
AU - Li, Siqing
AU - Ling, Leevan
N1 - Funding Information:
This work was partially supported by a Hong Kong Research Grant Council GRF Grant.
PY - 2019/8
Y1 - 2019/8
N2 - In this paper, we apply kernel-based collocation methods to elliptic problems with mixed boundary conditions. We propose some weighted least-squares formulations with different weights for the Dirichlet and Neumann boundary collocation terms. Besides fill distance of discrete sets, our weights also depend on other three factors: the proportion of measures of the Dirichlet and Neumann boundaries, dimensionless volume ratios of the boundary and domain, and kernel smoothness. We determine the dependencies of these terms in weights by different numerical tests. Our least-squares formulations can be proved convergent in H2(Ω). Numerical experiments for two dimensional examples show that we can obtain convergent solutions for kernel smoothness m ∈ {3, 4, 5} in the irregular domains, circle domain, and rectangle thin domain. We also apply our formulations to three dimensional cases and get desired convergent results for m ∈ {4, 5, 6, 7} in cubic, sphere and torus domain under different boundary conditions.
AB - In this paper, we apply kernel-based collocation methods to elliptic problems with mixed boundary conditions. We propose some weighted least-squares formulations with different weights for the Dirichlet and Neumann boundary collocation terms. Besides fill distance of discrete sets, our weights also depend on other three factors: the proportion of measures of the Dirichlet and Neumann boundaries, dimensionless volume ratios of the boundary and domain, and kernel smoothness. We determine the dependencies of these terms in weights by different numerical tests. Our least-squares formulations can be proved convergent in H2(Ω). Numerical experiments for two dimensional examples show that we can obtain convergent solutions for kernel smoothness m ∈ {3, 4, 5} in the irregular domains, circle domain, and rectangle thin domain. We also apply our formulations to three dimensional cases and get desired convergent results for m ∈ {4, 5, 6, 7} in cubic, sphere and torus domain under different boundary conditions.
KW - Elliptic partial differential equations
KW - Meshless method
KW - Mixed boundary conditions
KW - Weighted least-squares formulations
UR - http://www.scopus.com/inward/record.url?scp=85064815198&partnerID=8YFLogxK
U2 - 10.1016/j.enganabound.2019.04.012
DO - 10.1016/j.enganabound.2019.04.012
M3 - Journal article
AN - SCOPUS:85064815198
SN - 0955-7997
VL - 105
SP - 146
EP - 154
JO - Engineering Analysis with Boundary Elements
JF - Engineering Analysis with Boundary Elements
ER -