Galerkin method

Engineering

Differential equation

Applied mathematics

Stability (learning theory)

Weighting

Regularized meshless method

Computer science

Mathematical optimization

Collocation method

Radiology

Meshfree methods

Thermodynamics

Physics

Finite element method

Kernel (algebra)

Collocation (remote sensing)

Singular boundary method

Mathematics

Orthogonal collocation

Mathematical analysis

Boundary element method

Ordinary differential equation

Combinatorics

Structural engineering

Medicine

Moving least squares

Machine learning

Computer Methods in Applied Mechanics and Engineering

Meshfree direct collocation method (DCM) based on strong form suffers low accuracy and instability compared with Galerkin-based meshfree methods. Meshfree weighted least squares collocation method (WLSCM) can improve the accuracy and stability by minimizing the least squares functional corresponding to the governing equation and boundary conditions. However, usage of more discrete points in the least squares solution increases the computational cost which notably reduces the efficiency. In this paper, we propose a new meshfree stabilized collocation method (SCM) and introduce the reproducing kernel function as the approximation. Auxiliary collocation points located in the local subdomains are employed for the stabilization. Their proper positions and corresponding weighting factors for integration using constant and cubic B-spline weighting functions are derived to keep the consistency conditions. Therefore, in this method, the approximation can satisfy the consistency conditions not only on the points, but also in the subdomains. The suggested method has the comparable accuracy and stability compared with WLSCM, and the computation efficiency can match up to DCM. Numerical simulations demonstrate that SCM can surpass DCM in accuracy and stability, and outperform WLSCM in efficiency and conditioning of the matrix according to the discrete equations, where the condition number of matrix is related to the stability.

A meshfree stabilized collocation method (SCM) based on reproducing kernel approximation